A STUDY ON SOME VARIATIONS OF GRAPH LABELING AND ITS

By

M. SUBBIAH, M.Sc., M.Phil., PGDOR., M.C.A.,

(Reg No: 3625 / Ph.D / Maths / P.T. / April 2007)

Under the Guidance of

Dr. (Mrs.) B. GAYATHRI, M.Sc., M.Phil., Ph.D., P.G.D.C.A., M.C.A.,

A STUDY ON SOME VARIATIONS OF GRAPH

LABELING AND ITS APPLICATIONS

IN VARIOUS FIELDS

Thesis submitted to the Bharathidasan Universityin partial fulfilment of the requirements

for the degree ofDOCTOR OF PHILOSOPHY IN MATHEMATICS

PG & RESEARCH DEPARTMENT OF MATHEMATICS

TIRUCHIRAPPALLI - 620 023, INDIA

PERIYAR E.V.R. COLLEGE (Autonomous)

NOVEMBER 2011

Dr. (Mrs.) B. GAYATHRI,

M.Sc., M.Phil., Ph.D., P.G.D.C.A., M.C.A., Associate Professor of Mathematics,

PG and Research Department of Mathematics, Periyar E.V.R. College (Autonomous), Tiruchirappalli-620 023.

CERTIFICATE

This is to certify that M. SUBBIAH has prepared this thesis entitled

“A STUDY ON SOME VARIATIONS OF GRAPH LABELING AND

ITS APPLICATIONS IN VARIOUS FIELDS” submitted to

BHARATHIDASAN UNIVERSITY in partial fulfillment of the

requirements for the award of the degree of Doctor of Philosophy under my

guidance and supervision. This is a bonafide record of the independent and

original work done by him under my guidance in the PG and Research

Department of Mathematics, Periyar E.V.R. College (Autonomous),

Tiruchirappalli-23, from Apr. 2007 to Nov. 2011. This is also to certify that the

thesis represents his independent original work and has not previously formed

the basis for award of any degree, diploma, fellowship or any other similar title

in any institution or university.

(Dr. B. Gayathri)

Research Supervisor

M. SUBBIAH, H.O.D. of Mathematics

Department of Mathematics, VKS College of Engineering and Technology, Karur (Dt.)

DECLARATION

I, M. SUBBIAH, hereby declare that this thesis entitled “A STUDY

ON SOME VARIATIONS OF GRAPH LABELING AND ITS

APPLICATIONS IN VARIOUS FIELDS” submitted to the Bharathidasan

University in partial fulfillment of the requirements for the award of the degree

of Doctor of Philosophy, is a record of independent research work carried

out by me during the period for Apr. 2007 to Nov. 2011 under the supervision

of Dr. (Mrs.) B. Gayathri, Associate Professor, PG and Research Department

of Mathematics, Periyar E.V.R. College (Autonomous), Tiruchirappalli-23.

This work has not formed the basis for the award of any degree, diploma,

associate ship or fellowship, or any other similar title in any institution or

university to my knowledge.

(M. Subbiah) Research Scholar

Place: Tiruchirappalli – 620 023.

Date :

ACKNOWLEDGEMENT

I offer my thanks to Almighty GodAlmighty GodAlmighty GodAlmighty God for providing me with courage,

patience, good health and having enabled me to carry out these investigations.

With deep and profound sense of gratitude, I express my heartfelt and

sincere thanks to my research advisor, Dr. (Mrs.) B. Gayathri, Associate

Professor, PG and Research Department of Mathematics, Periyar E.V.R.

College (Autonomous), Tiruchirappalli-23, for her inspiring guidance, innovative

suggestions, kind cooperation and constant encouragement during the entire

period of work.

My sincere thanks are due to Prof. P. Gowthaman, M.Com.,

M.Phil., M.B.A., Principal, Periyar E.V.R. College (Autonomous)

Tiruchirapplli-23 for providing the necessary facilities throughout my research

work in this institution. I am grateful to Prof. P. Thamilmani, M.Sc.,

M.Phil., B.Ed., P.G.D.O.R., Head, PG and Research Department of

Mathematics and former Heads of the department for providing the facilities

and encouragement shown during the course of work. My sincere thanks are

due to the faculty members, department of Mathematics for their moral support

and my sincere thanks to the full - time research scholar of the department for

his timely help during the period of work.

I am grateful to the chairman Mr. K. Subburathinam, the Trust

members and the faculty members of VKS College of Engineering and

Technology for providing the facility and encouragement shown during the

course of work.

I express my grateful thanks to my doctoral committee member

Dr. M.A. Gopalan, Head, Department of Mathematics, Srimathi Indra

Gandhi College, Tiruchirappalli for his valuable suggestions.

I express my sincere thanks to M/s. Golden Net Computers,

Tiruchirappalli for neat type setting and binding my thesis.

With loving gratitude and heartfelt thanks, I would like to thank my

parents for their encouragement and sacrifice done to me.

I express my grateful thanks to my wife and daughters for their

constant support, encouragement and patience shown to me during the period

of work.

Subbiah. M

PUBLICATIONS

Journals & Proceedings

1. Gayathri. B and Subbiah. M, “Strong Edge − graceful Labeling For

Some Graphs”, Bulletin of Pure and Applied Sciences, Vol.1, 27E No.1,

(2008) pp. 1 -10.

2. Gayathri. B and Subbiah. M, “Edge − graceful Labeling of some trees”,

Global Journal of Mathematical Sciences: Theory and Practical, Vol. 3,

No. 1 (2011) pp. 27-33.

3. Gayathri. B and Subbiah. M, “Strong edge − graceful labeling of Kite

graphs” Global Journal of Theoretical and Applied Mathematics

Sciences, Vol. 1, No. 1 (2011) pp. 1-11.

4. Gayathri. B and Subbiah. M, “Strong edge − graceful labeling of some

trees”, Proceedings of the National Conference on Fuzzy Mathematics

and Graph Theory, Jamal Mohamed College, Trichy, Mar. 27-28, 2008,

pp. 208-213.

Papers presented and workshop attended


graphs” presented in the National Conference on Applications of

Mathematics, Periyar E.V.R. College, Trichy -23, March 22-23, 2007.

2. Gayathri. B and Subbiah. M, “Edge – graceful labeling of some trees”

presented in the National Conference on Recent Developments in

Mathematics and its Applications, Government Arts College for

Women, Pudukkottai, March 28-29, 2008.


trees” presented in the National Conference on Fuzzy Mathematics and

Graph Theory, Jamal Mohamed College, Trichy, Mar. 27-28, 2008.

4. Subbiah. M, participated in the Workshop on “Art of Scientific Article

Writing” held during March 11-12, 2011 in Periyar E.V.R. College,

Trichy -23.

CONTENTS

CHAPTER TITLE PAGE NO.

1. Introduction 1

2. Preliminaries 6

3. Strong Edge – Graceful Labeling of Graphs 10

4. Edge – Graceful Labeling of Some Trees 36

5. Some More Strong Edge – Graceful Graphs 52

6. Strong Edge – Graceful Labeling of Some Simple Graphs

87

7. Strong Edge – Graceful Labeling of Some Cycle Related Graphs

116

8. Strong Edge – Graceful Labeling of Some Disconnected Graphs

148

Bibliography 190

CHAPTER – I

Introduction

1

CHAPTER - I

INTRODUCTION

This work deals with graph labeling. All the graphs considered here are

finite and undirected. The terms not defined here are used in the sense of

Harary [10].

A graph labeling is an assignment of integers to the vertices or edges or

both subject to certain conditions. If the domain of the mapping is the set of

vertices (or edges) then the labeling is called a vertex labeling (or an edge

labeling).

By a (p, q) graph G, we mean a graph G = (V, E) with | V | = p and

| E | = q.

Graph labeling was first introduced in the late 1960's. Many studies in

graph labeling refer to Rosa's research in 1967 [19]. Rosa introduced a

function f from a set of vertices in a graph G to the set of integers {0, 1, 2,..., q}

so that each edge xy is assigned the label |f (x) – f (y)|, all labels are distinct.

Rosa called this labeling β �-valuation. Independently, Golomb [8] studied the

same type of labeling and called as graceful labeling.

Graceful labeling originated as a means of attacking the conjecture of

Ringel [18] that 2 1nK + can be decomposed into 2 1n + subgraphs that are all

isomorphic to a given tree with nedges. Those graphs that have some sort of

regularity of structure are said to be graceful. Sheppard [21] has shown that

there are exactly q! gracefully labeled graphs with q edges. Rosa [19]

suggested three possible reasons why a graph fails to be graceful.

2

1) G has “too many vertices” and “not enough edges”

2) G has “too many edges”

3) G has “wrong parity”

Labeled graphs serve as useful models for a broad range of applications

such as X-ray crystallography, radar, coding theory, astronomy, circuit design

and communication network addressing. Particularly, interesting applications

of graph labeling can be found in [1, 2, 3, 4, 5, 9, 20].

In the recent years, dozens of graph labeling techniques such as

α-labeling, k and ( , )k d -graceful, k-equitable, skolem graceful, odd graceful

and graceful like labeling have been studied in over 1000 papers [7].

In 1990, Antimagic graphs was introduced by Hartsfield and Ringel

[11]. A graph with q edges is said to be antimagic if its edges can be labeled

with 1, 2, …, q such that the sums of the labels of the edges incident to each

vertex are distinct. This notion of antimagic labeling was extended to

hypergraphs by Sonntag [24].

In 1985, Lo [17] introduced the edge – graceful graph which is a dual

notion of graceful labeling. A graph ( , )G V E is said to be edge – graceful if

there exists a bijection f from E to {1, 2, …, q } such that the induced

mapping f + from V to { }0, 1, 2, ..., 1p − given by ( )( )( ) (mod )f x f xy p+ = ∑

taken over all edges xy is a bijection.

Every edge – graceful graph is found to be antimagic and Lo [17] found

a necessary condition for a graph with p vertices and q edges to be edge –

graceful as ( )( 1)( 1) mod

2

p pq q p

++ ≡ .

3

Lee [14] noted that this necessary condition extends to any multigraph

with p vertices and q edges. He conjectured that any connected simple ( , )p q

graph with ( )( 1)( 1) mod

2

p pq q p

−+ ≡ is edge – graceful and found that this

condition is sufficient for the edge – gracefulness of connected graphs. He also

[15] conjectured that all trees of odd order are edge – graceful.

Small [23] has proved that spiders in which every vertex has odd degree

with the property that the distance from the vertex of degree greater than 2 to

each end vertex is the same, are edge – graceful. Keene and Simoson [12]

proved that all spiders are edge – graceful if the spider has odd order with

exactly three end vertices. Cabaniss, Low, and Mitchem [6] have shown that

regular spiders of odd order are edge – graceful. Lee, Seah, and Wang [16]

gave a complete characterization of edge – graceful knp graphs. Shiu, Lam,

and Cheng [22] proved that the composition of the path 3P and any null graph

of odd order is edge – graceful.

Lo proved that all odd cycles are edge – graceful. Wilson and Riskin

[25] proved that the Cartesian product of any number of odd cycles is edge –

graceful.

This thesis emerged with the growing interest in the notion of edge –

graceful graphs and its noteworthy conjectures.

By the necessary condition of edge – gracefulness of a graph one can

verify that even cycles, and paths of even length are not edge – graceful. But

whether trees of odd order are edge – graceful is still open.

Motivated by the notion of edge – graceful graphs and Lo’s conjecture,

we define a new type of labeling called strong edge – graceful labeling by

4

relaxing its range through which we can get strong edge – graceful labeling of

even order trees.

A ),( qp graph G is said to have strong edge – graceful labeling if

there exists an injection f from the edge set to 3

1,2, ...,2

q

so that the

induced mapping f + from the vertex set to { }0,1, ..., 2 1p − defined by

{ } }( ) ( ) | ( )f x f xy xy E G+ = ∈∑ (mod 2p ) are distinct. A graph G is said to be

strong edge – graceful if it admits a strong edge – graceful labeling.

In this thesis, we investigate strong edge – graceful labeling (SEGL) of

some graphs. In some situations, we also present edge – graceful labeling of

some trees towards attempting to the Lo’s Conjecture.

We now present chapter-wise summary.

Chapter I briefly introduces the thesis.

Chapter II provides the fundamental concepts of graph theory and

definitions of graphs which are needed for the rest of this thesis.

Chapter III defines a new type of labeling called strong edge – graceful

labeling by relaxing the range of edge – graceful labeling. Here, we discuss the

strong edge – graceful labeling of the graphs Fn, Cn, Pn, nC+ , K1,n, TW(n),

SP(13, m) and 1mP nK⊙ .

Chapter IV deals with the edge – graceful labeling schemes of the

bistar trees and some specific families of trees such as 1, 1,: ,n mK K Yn,

FC(1m, K1, n), mGn and 1, 1,:n nN K K .

Chapter V deals with the strong edge – graceful labeling of the graphs

1,@m nC K , M(Pn), Fl(n), ( )3tC and butterfly graphs.

5

Chapter VI deals with the strong edge – graceful labeling schemes of

the graphs Fln, , , , n n n nP M W T+ and Bn, m.

Chapter VII provides the strong edge – graceful labeling schemes of

the graphs 2 nK C⊙ , @n tC P , G(n), SF(n) and Sn.

Chapter VIII discusses the strong edge – graceful labeling schemes of

some disconnected graphs such as disjoint unions of paths, cycles and star

graphs.

Finally, the thesis ends with bibliography.

CHAPTER – II

Preliminaries

6

CHAPTER - II

PRELIMINARIES

Definition 2.1

A graph G = (V, E) is a finite non-empty set V of objects called vertices

together with a set E of unordered pairs of distinct vertices called edges.

Definition 2.2

The cardinality of the vertex set of graph G is called the order of G and

is denoted by p. The cardinality of its edge set is called the size of G and is

denoted by q. A graph with p vertices and q edges is called a (p, q) −−−− graph.

Definition 2.3

If e = (u, v) is an edge of G, we write e = uv and we say that u and v are

adjacent vertices in G. If two vertices are adjacent, then they are said to be

neighbours. Further, vertex u and edge e are said to be incident with each

other, as are v and e. If two distinct edges f and g are incident with a common

vertex, then f and g are said to be adjacent edges.

Definition 2.4

An edge having the same vertex as both its end vertices is called a self

loop or loop.

Definition 2.5

If there are more than one edge between the same given pair of vertices,

these edges are called parallel edges.

7

Definition 2.6

The number of edges incident on a vertex v with self − loops counted

twice is called the degree of the vertex v. It is denoted by d(v) or deg v.

The maximum degree of a graph G is denoted by ∆(G) and the

minimum degree of G is denoted by δ(G).

A vertex with degree 0 is called an isolated vertex.

A vertex with degree 1 is called a pendant vertex.

A graph is regular of degree r if every vertex of G has degree r. Such

graphs are called r − regular graphs.

Definition 2.7

A graph H is a subgraph of a graph G, if all the vertices and all the

edges of H are in G, and each edge of H has the same end vertices in G.

Definition 2.8

A spanning subgraph of G is a subgraph H with V(H) = V(G).

(i.e.) H and G having exactly the same vertex set.

Definition 2.9

A graph in which any two distinct vertices are adjacent is called a

complete graph and denoted by Kp.

Definition 2.10

Two graphs G and G' are said to be isomorphic, if there is a one to one

correspondence between their vertices and between their edges, such that the

incidence relationship is preserved.

8

Definition 2.11

A walk is a finite alternating sequence of vertices and edges, beginning

and ending with vertices such that each edge is incident with the vertices

proceeding and succeeding it [No edge appears more than once].

Definition 2.12

A walk which begins and ends with the same vertex is called a closed

walk. A walk that is not closed is called an open walk.

Definition 2.13

If all the edges of a walk are distinct, then it is called a trial. An open

walk in which no vertex appears more than once is called a path.

Definition 2.14

A walk in which no vertex (except the initial and the final vertices)

appears more than once is called a cycle.

Definition 2.15

A graph G is said to be connected if there is atleast one path between

every pair of vertices in G. Otherwise, G is disconnected. A disconnected

graph consists of two or more connected subgraphs. Each of these connected

subgraph is called a component.

Definition 2.16

A graph G is called bigraph (or) bipartite graph if V(G) can be

partitioned into two disjoint subsets V1, V2 such that every line of G joins a

points of V1 to a point of V2. (V1, V2) is called a bipartition of G.

9

Definition 2.17

A bipartite graph with bipartition (V1, V2) is called a complete bipartite

graph if every vertex of V1 is adjacent to every vertex of V2.

Definition 2.18

A graph with exactly one cycle is called a unicyclic graph.

CHAPTER – III

Strong edge – graceful labeling of Graphs

10

CHAPTER – III

STRONG EDGE – GRACEFUL LABELING

OF GRAPHS

3.1. Introduction

Lo [17] introduced the notion of edge − graceful graphs. The necessary

condition for a graph (p, q) to be edge-graceful is q(q+1) ≡ 0 or ( )mod 2

pp .

With this condition, even cycles and paths of even length are not edge − graceful.

But whether trees of odd order are edge − graceful is still open. On these lines,

we define a new type of labeling called strong edge − graceful labeling by

relaxing its range in which almost all even order trees are found to be strong

edge − graceful.

So, in this chapter, we have introduced the strong edge − graceful

labeling of graphs and established the strong edge − gracefulness of fan, twig,

path, cycle, star, crown, spider and Pm�nK1.

Definition 3.1.1

A graph G with q edges and p vertices is said to have an edge −−−− graceful

labeling (EGL) if there exists a bijection f from the edge set to the set {1, 2, ..., q}

so that the induced mapping f + from the vertex set to the set {0, 1, 2,..., p – 1}

given by ( )f x+ = { }( ) | ( )f xy xy E G∈∑ (mod p) is a bijection. A graph G is

edge −−−− graceful graph (EGG) if it admits a edge − graceful labeling.

11

Definition 3.1.2

A (p, q) graph G is said to have strong edge −−−− graceful labeling (SEGL)

if there exists an injection f from the edge set to 3

1, 2, ..., 2

q

so that the

induced mapping f + from the vertex set to {0, 1, ..., 2p – 1} defined by

( )f x+ = ( ) ( ){ }|f xy xy E G∈∑ (mod 2p) are distinct. A graph G is said to be

strong edge – graceful graph (SEGG) if it admits a strong edge − graceful

labeling.

For the clear understanding of the above definitions we illustrate the

following examples.

Example 3.1.3

Consider the graph G = P4. Here q(q + 1) = 3 × 4 = 12 ≡ 0 (mod 4).

Thus, Lo’s necessary condition is satisfied. But it is not edge graceful.

Because if 3 is assigned to a pendant edge then the labels 1 and 2 cannot be

adjacent. Also if 1 is assigned to a pendant edge then the labels 2 and 3 cannot

be adjacent.

Again if 2 is assigned to a pendant edge then either 1 or 3 has to be the

label of another pendant edge which is impossible.

But the strong edge − graceful labeling of P4 is given below.

1 3 6 4

1 2 4

Thus, P4 is strong edge − graceful but not edge − graceful.

12

Example 3.1.4

The graph given in this example is both edge − graceful and strong

edge − graceful.

11

6

6

4 4 0

3

3

55

2

2

Herep = 7q = 6

mod p = 7

Fig. 3.1: EGL

22

9

9

4 7 10

8

8

66

3

3

Herep = 7q = 6

mod 2p = 14

Fig. 3.2: SEGL

Remark 3.1.5

All edge-graceful (p, q) graphs turn to be strong edge − graceful and we

observe that almost all graphs are strong edge – graceful due to the relaxed

range of values. So, we focus on finding the appropriate strong edge – graceful

labeling schemes of some classes of graphs.

3.2. Fan graph Fn

Definition 3.2.1

Let Pn denote the path on n vertices. Then the join of K1 with Pn is

defined as fan and is denoted by Fn (i.e) Fn = K1 + Pn.

13

Theorem 3.2.2

The fan F4n−2 (n ≥ 2) is strong edge − graceful for all n.

Proof

Let {v, v1, v2, v3, ..., vt} be the vertices of Ft and {vvi : 1 ≤ i ≤ t} ∪

{ ei = (vi, vi+1) : 1 ≤ i ≤ t – 1} be the edges of Ft which are denoted as in Fig. 3.3.

v1 v2 v3vt

e1 e2 e3

v

e2t-1

et-1

etet+1

... vt-1

Fig. 3.3: Ft with ordinary labeling

Take t = 4n – 2.

We first label the edges of F4n−2 as follows:

( )if e = i 1 ≤ i ≤ 8n – 5

Then the induced vertex labels are

( )if v+ = 2(4n – 2) + i – 1 1 ≤ i ≤ 2

( )3f v+ = 0

( )if v+ = i – 3 4 ≤ i ≤ 4n – 3

( )4 2nf v+− = 8n – 5 ; ( )f v+ = 4n + 1

Clearly, the vertex labels are all distinct. Hence, the fan F4n−2 (n ≥ 2) is

strong edge − graceful for all n.

14

The SEGL of F6, F10 are illustrated in Fig. 3.4 and Fig. 3.5 respectively.

9

12 13 0 1 2 11

1110

9 8 7 6

1 2 3 4 5

20 21 0 1 2 3 4 5 6 19

1918 17 16 15 14 13 12 11 10

13

1 2 3 4 5 6 7 8 9

Fig. 3.4: F6 with SEGL Fig. 3.5: F10 with SEGL

Theorem 3.2.3

The fan F4n+1 (n ≥ 1) is strong edge − graceful for all n.

Proof


{ ei = (vi, vi+1) : 1 ≤ i ≤ t – 1} be the edges of Ft which are denoted as in Fig. 3.3,

of Theorem 3.2.2.

Take t = 4n + 1.

We first label the edges of F4n+1 as follows:

( )if e =

4

1 4 1 8

4 1 8 1

i i n

i n i n

n i n

1 ≤ ≤ + + ≤ ≤ + = +

Then the induced vertex labels are:

( )1f v+ = 4n + 2

( )if v+ = n – 2 2 ≤ i ≤ 4n

( )4 1nf v++ = 8n + 2 ; ( )f v+ = 6n + 5

15

Clearly, the vertex labels are all distinct. Hence, the fan F4n+1 (n ≥ 1) is



5

11

9 8 7 6

6 0 1 2 10

1 2 3 4

17

9 1716 1514 13 121110

1 2 3 4 5 6 7

10 0 1 2 3 4 5 68

18


Theorem 3.2.4

The fan F4n (n ≥ 1) is strong edge − graceful for all n.

Proof



of Theorem 3.2.2.

Take t = 4n.

We first label the edges of F4n as follows:

( )if e = i 1 ≤ i ≤ 4n – 1

( )if e = i +1 4n ≤ i ≤ 8n – 1


( )f v+ = 4n + 2 ; ( )1f v+ = 8n + 1

( )1if v++ = i – 1 1 ≤ i ≤ 4n – 2

( )4nf v+ = 8n

16

Clearly, the vertex labels are all distinct. Hence, the fan F4n

(n ≥ 1) is strong edge − graceful for all n.

The SEGL of F16 is illustrated in Fig. 3.8.

18

32 31 30 2928 27 26 25 2423 22 21 20 1918 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

33 0 1 2 3 4 5 6 7 8 9 10 11 12 13 32

Fig. 3.8: F16 with SEGL

Theorem 3.2.5

The fan F4n−1 (n ≥ 2) is strong edge − graceful for all n.

Proof



of Theorem 3.2.2.

Take t = 4n – 1.

We first label the edges of F4n-1 as follows:

( )if e = i 1 ≤ i ≤ 8n – 3


( )f v+ = 4 4

2

n +

17

( )if v+ = 8n – 3 + i 1 ≤ i ≤ 2

( )if v+ = i – 3 3 ≤ i ≤ 4n – 2

( )4 1nf v+− = 8n – 3

Clearly, the vertex labels are all distinct. Hence, the fan F4n−1 (n ≥ 2) is



14 15 0 1 2 3 13

1 2 3 4 5 6

13 12 11 10 9 8 7

6

10

29 28 27 26 25 24 23 22 21 20 1918 17 16 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14

30 31 0 1 2 3 4 5 6 7 8 9 10 11 29


3.3. Cycle graph Cn

Theorem 3.3.1

Cn (n ≥ 3) is strong edge − graceful for all n when n is odd.

Proof

Let {v1, v2, v3, ..., vn} be the vertices of Cn and {e1, e2, e3, ..., en} be the

edges of Cn which are denoted as in Fig 3.11.

18

vn-1

en-1

vn

en

v1 e1 v2

e2

v3

e3

v4

...

......

Fig. 3.11: Cn with ordinary labeling

We first label the edges of Cn as follows:

( )if e = i 1 ≤ i ≤ n


( )1f v+ = n + 1

( )if v+ = 2i – 1 2 ≤ i ≤ n

Clearly, the vertex labels are all distinct. Hence, Cn (n ≥ 3) is strong

edge − graceful for all n when n is odd.

The SEGL of C7 and C11 are illustrated in Fig. 3.12 and Fig. 3.13

respectively.

9

7

5

38

13

11

5

6

7

1

2

3

4

13

11

9

7

5

312

21

19

17

157

8

9

10

111

2

3

4

5

6

Fig. 3.12: C7 with SEGL Fig. 3.13: C11 with SEGL

19

Theorem 3.3.2

Cn (n ≥ 3) is strong edge − graceful for all n when n is even.

Proof

Let {v1, v2, v3, ..., vn} be the vertices of Cn and {e1, e2, e3, ..., en} be the

edges of Cn which are denoted as in Fig 3.11, of Theorem 3.3.1.

We first label the edges of Cn as follows:

( )if e = i 1 ≤ i ≤ n – 1

( )nf e = n + 1


( )1f v+ = n + 2

( )if v+ = 2i – 1 2≤ i ≤ n – 1

( )nf v+ = 0

Clearly, the vertex labels are all distinct. Hence, Cn (n ≥ 3) is strong

edge − graceful for all n when n is even.

The SEGL of C6 and C8 are illustrated in Fig. 3.14 and Fig. 3.15

respectively.

7

5

38

0

9 4

35

7

1

2

11 9

7

5

310

0

13

6

5

4

3

2

1

9

7

Fig. 3.14: C6 with SEGL Fig. 3.15: C8 with SEGL

20

3.4. Path graph Pn

Theorem 3.4.1

Pn (n ≥ 3) is strong edge − graceful for all n odd.

Proof

Let {v1, v2, v3, ..., vn} be the vertices of Pn and {e1, e2, e3, ..., en-1} be the

edges of Pn which are denoted as in Fig. 3.16.

v1 v2 v3

e1 e2 e3 en-1

vn-1 vn

en-2. . .

Fig. 3.16: Pn with ordinary labeling

We first label the edges of Pn as follows:

( )if e = i 1 ≤ i ≤ n – 1


( )1f v+ = 1

( )if v+ = 2i – 1 2 ≤ i ≤ n – 1

( )nf v+ = n – 1

Clearly, the vertex labels are all distinct. Hence, Pn (n ≥ 3) is strong

edge − graceful for all n odd.

The SEGL of P9 and P13 are illustrated Fig. 3.17 and Fig. 3.18

respectively.

1 3 5 7 9 11 13 15 8

1 2 3 4 5 6 7 8

Fig. 3.17: P9 with SEGL

1 3 5 7 9 11 13 15 17

1 2 3 4 5 6 7 8

19 21 23 12

9 10 11 12


21

Theorem 3.4.2

Pn (n ≥ 3) is strong edge − graceful for all n even.

Proof

Let {v1, v2, v3, ..., vn} be the vertices of Pn and {e1, e2, e3, ..., en−1} be the

edges of Pn which are denoted as in Fig. 3.16, of Theorem 3.4.1.

We first label the edges of Pn as follows:

( )if e = i 1 ≤ i ≤ n – 2

( )1nf e − = n


( )if v+ = 2i – 1 1 ≤ i ≤ n – 2

( )1nf v+− = 2n – 2 ; ( )nf v+ = n

Clearly, the vertex labels are all distinct. Hence, Pn (n ≥ 3) is strong

edge − graceful for all n even.

The SEGL of P8 and P12 are illustrated in Fig. 3.19 and Fig. 3.20

respectively.

1 3 5

1 2 3 8

8

6

7 9 11 14

4 5


1 3 5 7 9 11 13 15 17

1 2 3 4 5 6 7 8

19 22 12

9 10 12


22

3.5. Crown graph +nC

Theorem 3.5.1

nC+ (n ≥ 3) is strong edge − graceful for all n.

Proof

Let {v1, v2, ..., vn} ∪ { }' ' '1 2, , ..., nv v v be the vertices of nC+ and

{ ei = vi vi+1 : 1 ≤ i ≤ n – 1} ∪ {en = vnv1} ∪ { 'i iv v : 1 ≤ i ≤ n} be the edges of nC+

which are denoted as in Fig. 3.21.

en-1

vn

vn-1

en

v1

e1

v2

e2

v3

e2'

'1v

'nv

'1e

'ne

'1ne −

'1nv −

'2v

..

.

.

.

.

..

.

...

..

.

Fig. 3.21: +

nC with ordinary labeling

We first label the edges of nC+ as follows:

( )if e = i 1 ≤ i ≤ n

( )'i if v v = 2n – i + 1 1 ≤ i ≤ n


( )if v = 2n + 1 + i 1 ≤ i ≤ n

( )'if v = 2n + 1 – i 1 ≤ i ≤ n

Clearly, the vertex labels are all distinct. Hence, nC+ (n ≥ 3) is strong

edge − graceful for all n.

23

The SEGL of 7C+ and 8C+ are illustrated in Fig. 3.22 and Fig. 3.23

respectively.

10

20

11

12

1314

8

9

10

11

12

13

14

8

9

3

4

2

5

6

7

1

19

18

17

1622

21

12

13

14

1516

9

10

11

1211

10

9

16 15

14

13

5

4

3

2

1

8

7

6

20

19

1825

24

23

22 21

Fig. 3.22: +7C with SEGL Fig. 3.23: +

8C with SEGL

3.6. Star graph K1, n

Theorem 3.6.1

K1, 2n (n ≥ 2) is strong edge − graceful for all n.

Proof

Let {v, v1, v2, ..., vt} be the vertices of K1, t and {e1, e2, e3, ..., et} be the

edges of K1, t which are denoted as in Fig. 3.24.

v

v1 v2 v3 vt-1 vt

et-1 ete1 e2 e3

...

Fig. 3.24: K1, t with ordinary labeling

24

Take t = 2n.

We first label the edges of K1,2n as follows

( )if e = i 1 ≤ i ≤ 2n


( )if v+ = i 1 ≤ i ≤ 2n

( )f v+ = 2 1 if is odd

0 if is even

n n

n

+

Clearly, the vertex labels are all distinct. Hence, K1, 2n (n ≥ 2) is strong


The SEGL of K1,6 and K1,8 are illustrated in Fig. 3.25 and Fig 3.26

respectively.

7

1 2 3 4 5 6

12 3 4 5

6

0

1 2 3 4 5 6

12 3 4 5 6

7 8

7 8

Fig. 3.25: K1, 6 with SEGL Fig. 3.26: K1, 8 with SEGL Theorem 3.6.2

K1, 4n−1 (n ≥ 1) is strong edge − graceful for all n.

Proof


edges of K1, t which are denoted as in Fig. 3.24, of Theorem 3.6.1.

Take t = 4n – 1.

25

We first label the edges of K1,4n-1 as follows:

( )if e = i 1 ≤ i ≤ 4n – 1


( )if v+ = i 1 ≤ i ≤ 4n – 1

( )f v+ = 6n

Clearly, the vertex labels are all distinct. Hence, K1, 4n−1 (n ≥ 1) is strong


The SEGL of K1,7 and K1,11 are illustrated in Fig. 3.27 and Fig 3.28

respectively.

12

1 2 3 4 5 6

12 3 4 5 6

7

7

18

1 2 3 4 5 6

12 3 4 5 6

7 8

7 8

9 10 11

9 10 11

Fig. 3.27: K1, 7 with SEGL Fig. 3.28: K1, 11 with SEGL

Theorem 3.6.3

K1, 4n+1 (n ≥ 1) is strong edge − graceful for all n.

Proof


edges of K1, t which are denoted as in Fig. 3.24, of Theorem 3.6.1.

Take t = 4n + 1.

We first label the edges of K1, 4n+1 as follows:

( )if e = i 1 ≤ i ≤ 4n

( )4 1nf e + = 3

2

q

26


( )if v+ = i 1 ≤ i ≤ 4n

( )4 1nf v++ =

3

2

q

; ( )f v+ = 4n + 1

Clearly, the vertex labels are all distinct. Hence, K1, 4n+1 (n ≥ 1) is strong


The SEGL of K1,5 and K1,9 are illustrated in Fig. 3.29 and Fig. 3.30

respectively.

5

1 2 3 4 7

1 2 3 4 7

9

1 2 3 4 5

1 2 3 4 5

6 7

6 7

8 13

8 13

Fig. 3.29: K1, 5 with SEGL Fig. 3.30: K1, 9 with SEGL

3.7. Twig graph TW(n)

Theorem 3.7.1

The twig graph TW(n) (n ≥ 4) is strong edge − graceful for all n.

Proof

Let {v1, v2, v3, ..., vn–1, vn, vn+1, ..., v3n–4} be the vertices and

{ e1, e2, e3, ..., en-1, en, en+1, ..., e2n–3, e2n–2, ..., e3n–5} be the edges of the twig

graph TW(n) which are denoted as in Fig. 3.31.

v2n-3 vn+2

e2n-3

v1v2 v3 v4 v5 vne1 e2 e3 e4

e2n-2 e3n-5

v2n-2 v3n-5

en-2

vn+1

v2n-1

e2n-1

...

v3n-6

en-1

e3n-6

enen+1

vn-1

v2n-4

e2n-4

... ...

Fig. 3.31: TW(n) with ordinary labeling

27

We first label the edges as follows:

( )if e = i 1 ≤ i ≤ 3n – 5


( )1f v+ = 1

( )1if v++ = 4(n – 1) + 2i 1 ≤ i ≤ n – 3

( )1nf v+− = 0

( )if v+ = i – 1 n ≤ i ≤ 3n – 4

Clearly, the vertex labels are all distinct. Hence, the twig graph TW(n)

(n ≥ 4) is strong edge − graceful for all n.

The SEGL of TW(11) and TW(8) are illustrated in Fig. 3.32 and

Fig. 3.33 respectively.

19 18 17 16 15 14 13 12 11

19 18 17 16 15 14 13 12 11

42 44 46 48 50 52 54 56 01 101 2 3 4 5 6 7 8 9 10

20 21 22 23 24 25 26 27 28

20 21 22 23 24 25 26 27 28

Fig. 3.32: TW(11) with SEGL

1 30 32 34 36 38 0 7

1 2 3 4 5 6 7

13 12 11 10 9 8

13 12 11 10 9 8

14 15 16 17 18 19

191817161514

Fig. 3.33: TW(8) with SEGL

28

3.8. Spider graph

Theorem 3.8.1

The spider graph SP(13, m) is strong edge − graceful when m is even.

Proof

Let {v0, v1, v2, v3, ..., v3m} be the vertices of SP(13, m) and

{ e1, e2, e3, ..., em, em+1, ..., e2m, e2m+1, ..., e3m-1, e3m} be the edges of SP(13, m)

which are denoted as in Fig. 3.34.

v2m

v2m-1 v2m-2

vm+2

vm+1 v0

vm

v3m-2 v3m

v3m-1

v4v3

v2 v1

e2m

e2m-1

em+3

em+2

em+1

e 2m+1

em

e2m

+2

e 2m+3 e3m

e4

e3

e2

e1

v2m+1

v2m+2

..

.

..

.

...

Fig. 3.34: SP(13, m) with ordinary labeling


( )if e = i 1 ≤ i ≤ 3m


( )if v+ = 2i – 1 1 ≤ i ≤ m

( )if v+ = 2i + 1 m + 1 ≤ i ≤ 2m – 1

29

( )if v+ = i i = 2m

( )if v+ = 2i + 1 2m + 1 ≤ i ≤ 3m – 1

( )if v+ = i i = 3m

( )0f v+ = 4m + 2

Clearly, the vertex labels are all distinct. Hence, the spider graph

SP(13, m) is strong edge − graceful when m is even.

The SEGL of SP(13, 8) and SP(13, 12) are illustrated in Fig. 3.35 and


16

16

3115

29

14

2713

25

23

11 21

10

19

9

34

8

157

13

6

115

9

4

7

35

2

3 1 1

17

35

18 19

39

20 21

43

22 23

47

24

2437 41 45

12

Fig. 3.35: SP(13, 8) with SEGL

30

24

24

47

2345

22

43

2141

20

39

19 37

18

35

1733

16

31

15 29

14

27

13

50

12

2311

21

10

19

917

8

15

7 13

6

11

59

4

7

35

2

3

11

25

51

26

53

27

55

28

57

29

59

30

61

31

63

32

65

33

67

34

69

35

71

36

36


Theorem 3.8.2

The spider graph SP(13, m) is strong edge − graceful when m is odd.

Proof

Let {v1, v2, v3, ..., v3m} be the vertices of SP(13, m) and {e1, e2, e3, ..., em,

em+1, ..., e2m, e2m+1, ..., e3m-1, e3m} be the edges of SP(13, m) which are denoted as

in Fig 3.34, of Theorem 3.8.1.


( )if e = i 1 ≤ i ≤ 3m – 1

( )if e = i + 1 i = 3m


( )if v+ = 2i – 1 1 ≤ i ≤ m

31

( )if v+ = 2i + 1 m + 1 ≤ i ≤ 2m – 1

( )if v+ = i i = 2m

( )if v+ = 2i + 1 2m + 1 ≤ i ≤ 3m – 2

( )if v+ = 2i + 2 i = 3m – 1

( )if v+ = i + 1 i = 3m

( )0f v+ = 4m + 2

Clearly, the vertex labels are all distinct. Hence, the spider graph

SP(13, m) is strong edge − graceful when m is odd.

The SEGL of SP(13 ,7) and SP(13, 11) are illustrated in Fig. 3.37 and


1414

27

13

25

1223

11

21

10 19

9

17

8

30

7

13

611

5

9

47

3

5

23

1

1

15

31

16

33

17

35

18

37

19

39

20

42

22

22


32

2243

22

21

41

2039

19

37

18 35

17

33

1631

15

29

14 27

13

25

12

46

11

2110

19

9

17

815

7

13

6 11

5

9

47

3

5

23

1

1

23

47

24

49

25

51

26

53

27

55

28

57

29

59

30

61

31

63

32

66

34

34


3.9. Pm �� nK1 Graph

Theorem 3.9.1

The graph Pm � nK1 is strong edge − graceful when m and n are even.

Proof

Let V(Pm � nK1) = {v1, v2,...,vm, u11,...,u1n, u21, u22,...,u2n,...,um1, um2,...,umn}

and E(Pm � nK1) = {(vi, uij) : 1 ≤ i ≤ m, 1 ≤ j ≤ n ; (vi, vi+1) : 1 ≤ i ≤ m – 1} be the

vertices and edges of Pm � nK1 which are denoted as in Fig. 3.39.

v1 v2 vm-1 vm

u11 u12 u1nu21 u22 u2n

um1 um2 umn

...

... ...... ...

Fig. 3.39: Pm �� nK1 with ordinary labeling

33

We first label the edges of Pm� nK1 as follows:

( )1,i if v v+ = i 1 ≤ i ≤ m – 2

( )1,m mf v v− = m

Now consider the Diophantine equation x1 + x2 = 2p. Consider the

distinct pairs of solution for this equation of the form (t, 2p − t) where

2p − 3

2

q

≤ t ≤ 3

2

q

. Assign any of these pairs to the pendant edges in any

order pairwise.


( )if v+ = 2i – 1 1 ≤ i ≤ m – 2

( )if v+ = 2(m – 1) i = m – 1

( )if v+ = m i = m

The pendant vertices will receive the labels of the edges with which they

are incident.

Clearly, the vertex labels are all distinct. Hence, the graph Pm � nK1 is

strong edge − graceful when m and n are even.

The SEGL of P6 � 6K1 is illustrated as in Fig. 3.40.

1

41 43 40 44 3945 38 46 37 47 36

48 35 49 34 50 3351 32 52 31 53 30

54 29 55 28 56 2757 26

58 25 59 2460

1 3 2 5 3 7 4 10 6 6

41 43 40 44 39 45 38 46 37 47 36 48 35 49 34 50 33 51 32 52 31 53 30 54 29 55 28 56 27 57 26 58 25 59 24 60

Fig. 3.40: P6 �� 6K1 with SEGL

34

Theorem 3.9.2

The graph Pm � nK1 is strong edge – graceful when m is odd and n is

even.

Proof

Let V(Pm � nK1) = {v1, v2,...,vm, u11,...,u1n, u21, u22,...,u2n,...,um1, um2,...,umn}

and E(Pm � nK1) = {(vi, uij) : 1 ≤ i ≤ m, 1 ≤ j ≤ n ; (vi, vi+1) : 1 ≤ i ≤ m – 1} be the

vertices and edges of Pm � nK1 which are denoted as in Fig. 3.39, of

Theorem 3.9.1.

We first label the edges of Pm � nK1 as follows:

( )1,i if v v+ = i 1 ≤ i ≤ m – 1

Now consider the Diophantine equation x1 + x2 = 2p. Consider the

distinct pairs of solution for this equation of the form (t, 2p – t) where

2p − 3

2

q

≤ t ≤ 3

2

q

. Assign any of these pairs to the pendant edges in any

order pairwise.


( )if v+ = 2i – 1 1 ≤ i ≤ m – 1

( )mf v+ = m – 1

The pendant vertices will receive the labels of the edges with which they

are incident.

Clearly, the vertex labels are all distinct. Hence, the graph Pm � nK1 is

strong edge – graceful when m is odd and n is even.

35

The SEGL of P5 � 4K1 and P7 � 6K1 are illustrated as in Fig. 3.41 and


1 1 3 2 5 3 7 4 4

36 14 35 15

36 14 35 15

34 16 33 17

34 16 33 17

32 18 3119

32 18 31 19

30 20 21

30 20 29

29

21

28 22 27 23

28 22 27 23


6 61 1 3 2 5 3 7 4 9 5 11

4850 47 51 46 52

48 50 47 51 4652

4553 44 54 43 55

45 53 44 54 4355

4256 41 57 40 58

42 56 41 57 4058

3959 38 60 37 61

39 59 38 60 3761

36 62 35 63 34 64

36 62 35 63 3464

33 65 32 66 31 67

33 65 32 66 3167

3068 29 69 28 70

3068 29 69 28

70


CHAPTER – IV

Edge – graceful Labeling of Some Trees

36

CHAPTER - IV

EDGE – GRACEFUL LABELING OF SOME TREES

4.1. Introduction

In Chapter – III, we introduced strong edge – graceful labeling and

established strong edge – graceful labeling of certain families of graphs. In this

chapter, we discuss edge – graceful labeling of some trees such as

� 1, 1,:n mK K .

� 1, 1,:n mN K K .

� Yn.

� FC (1m, K1, n).

� mGn.

4.2. Edge – graceful labeling

Definition 4.2.1 [17]

A graph G(V, E) is said to be edge – graceful if there exists a bijection

f from E to {1, 2, ..., q} such that the induced mapping f + from V to

{0, 1, 2, ..., p−1} given by ( )f x+ = ( )( )( )modf xy p∑ taken over all edges

xy is a bijection.

Lo [17] found a necessary condition for a graph with p vertices and

q edges to be edge – graceful as q(q + 1) ≡ 0 or 2

p (mod p).

Definition 4.2.2

The graph 1, 1,:n mK K is obtained by joining the center u of the star K1, n

and the center v of another star K1, m to a new vertex w. The number of vertices

is n + m + 3 and the number of edges is n + m + 2.

37

Theorem 4.2.3

The graph 1, 1,:n mK K is edge – graceful for all n, m even and n ≠ m.

Proof

Let {w, v, v', v1, v2, ..., vn, vn+1, ..., vn+m} be the vertices of 1, 1,:n mK K

and {e1, e2, ..., en+m} ∪ {e, e'} be the edges of 1, 1,:n mK K which are denoted as

in Fig. 4.1.

ev v'w

e'

vn

vn-1

vn-2

v2

v1

en

en-1

en-2

e2

e1

vn+m

vn+m-1

vn+2

vn+1

en+m

en+m-1

en+2en+1

Fig. 4.1: 1, 1,:n mK K with ordinary labeling


Consider the Diophantine equation x1 + x2 = p. The solutions are of the

form (t, p – t) where 1 ≤ t ≤ 2

q. There will be

2

q pair of solutions.

We first define ( )f e = q ; ( )'f e = 1.

From other pairs, we label the edges of K1,n and K1,m by the coordinates

of the pair in any order so that adjacent edges receive the coordinates of the pairs.


( )f w+ = 0 ; ( )'f v+ = 1 ; ( )f v+ = q

Now, the pendant vertices will have labels of the edges with which they

are incident and they are distinct. Hence, the graph 1, 1,:n mK K is edge –

graceful for all n, m even and n ≠ m.

38

The EGL of 1,4 1,2:K K is illustrated in Fig. 4.2.

2

7

6

3

4

5

2

7

6

3

4

58

8

0

11

Fig. 4.2: B4,2 with EGL

Theorem 4.2.4

The graph 1, 1,:n mK K is edge – graceful for all n, m odd and n ≠ m.

Proof



in Fig. 4.1, of Theorem 4.2.3.



q. There will be

2


Among 2

q pair of solutions, choose the pair (q, 1).

We now label the edges as follows:

f (e) = q ; f (e') = 1 ; f (e1) = 2 ; ( )1nf e + = p – 2

As we have labeled the edges e1 and en+1 there will be even number of

edges incident with each v and v' to be labeled. We label these edges as was

done in Theorem 4.2.3.


( )f w+ = 0 ; ( )f v+ = 1 ; ( )'f v+ = p – 1

39

The pendant vertices will now receive the labels of the edges with which

they are incident and they are distinct. Hence, the graph 1, 1,:n mK K is edge –

graceful for all n, m odd and n ≠ m.


33

8 8

2

2

1

100

1

10

4

47

7

55

6

6

9

9

Fig. 4.3: 1, 1,:3 5K K with EGL

Theorem 4.2.5

The graph 1, 1,:n mK K is edge – graceful for all n is even and m = n.

Proof




Consider the Diophantine equation, x1 + x2 = p. The solutions are of the


q. There will be

2


We first label the edges e and e1 as follows:

( )f e = q ; ( )'f e = 1

40

From the remaining pairs, label the edges of K1,n and K1,m by the

coordinates of the pair in any order so that adjacent edges receives the

coordinates of the pairs.


( )f w+ = 0 ; ( )'f v+ = 1 ; ( )f v+ = q

Now the pendant vertices will receive labels of the edges with which

they are incident and they are distinct. Hence, the graph 1, 1,:n mK K is edge –

graceful for all n is even.


10 10

10 1

83

2

8

3

2

4

7

6

6

4

7

99

5

5

Fig. 4.4: 1,4 1,4:K K with EGL

Theorem 4.2.6

The graph 1, 1,:n nK K is edge – graceful for all n is odd.

Proof





form (t, p – t) where1 ≤ t ≤ q/2. There will be q/2 pair of solutions.

41


( )f e = q ; ( )'f e = 1

( )1f e = 2 ; ( )1nf e + = p – 2

As we have labeled the edges e1 and en+1, there will be even number of

edges incident with each v and v1 to be labeled. We label these edges as was

done in Theorem 4.2.3.


( )f w+ = 0 ; ( )f v+ = 1 ; ( )'f v+ = p – 1

The pendant vertices will have the labels of the edges with which they

are incident and they are distinct. Hence, the graph 1, 1,:n nK K is edge –

graceful for all n is odd.


33

6 6

2

2

1

80

1

8

44

55

7

7

Fig. 4.5: 1, 1,:3 3K K with EGL

Theorem 4.2.7

The graph 1, 1,:n mK K of even order is not an edge – graceful graph.

Proof

The graph 1, 1,:n mK K will be of even order in the following cases.

(i) n is odd and m is even

(ii) n is even and m is odd

42

In both cases, p = n + m + 3 which is even and hence by Lo’s necessary

condition, it is not edge − graceful.

Definition 4.2.8

The graph 1, 1,:n mN K K is obtained by joining the each center of N

copies of 1, 1,:n mK K to a new vertex w.

Theorem 4.2.9

The graph 1, 1,:n mN K K (n, m ≥ 2) is edge – graceful for all N, n, m is

even.

Proof

Let {ui, vi, wi : 1 ≤ i ≤ N} ∪ {w0} ∪ {uij : 1 ≤ i ≤ N, 1 ≤ j ≤ n}

∪ {vi,j : 1 ≤ i ≤ N, 1 ≤ j ≤ m} be the vertices

and {ai1, ai2 : 1 ≤ i ≤ N} ∪ {ei : 1 ≤ i ≤ N} ∪ {eij : 1 ≤ i ≤ N, 1 ≤ j ≤ n} ∪

{ gij : 1 ≤ j ≤ N, 1 ≤ j ≤ m} be the edges of 1, 1,:n mN K K which are denoted as

in Fig. 4.6.

Consider the Diophantine equation x1 + x2 = p. The pair of solutions are

(t, p – t) where 1 ≤ t ≤ 2

q. There are

2

q pair of solutions. With these pairs, we

label the edges by the co-ordinates of the pair in any order so that the adjacent

edges receive the coordinates of the pairs. Now the pendant vertices will have

the labels of the edges with which they are incident and they are distinct.

Now the induced vertex labels are follows:

( )if w+ = wi 1 ≤ i ≤ N ; ( )0f w+ = 0.

( )if w+ = ( )if e 1 ≤ i ≤ N

( )if u+ = ( )1if a+ 1 ≤ i ≤ N

43

( )if v+ = ( )2if a+ 1 ≤ i ≤ N

Hence, the graph 1, 1,:n mN K K is (n, m ≥ 2) is edge – graceful for all

N, n, m is even.

v2m

v22

v21

..

.

v2a22

w2

a21

u2

g2m

g22

g21

..

.

u2n

u22u21

.

.

.

e2n

e22e21

.

.

.

v1

v1m

v12

v11

a12

a11

w1

g1m

g12g11

.

.

.

u1n

u12

u11

...

e1ne12

e11

u1

vNm

vN1

vN2

...

gNm

gN2

gN1

vNaN2

aN1 wN

uN

uNn

uN1

uN2

.

.

.

eNn

eN2

eN1

.

.

.

...

.

.

.

.

.

.

...

v3m

v32

v31

.

.

.

g3m

g32

g31

.

.

.

v3

a32 w3

u3a31

u3n u32

u31

e3n e32

e31

...

...

e3 e2

e1eN

w0

Fig. 4.6: 1, 1,:n mN K K with ordinary labeling

44

The EGL of 1,2 1,24 :K K and 1,2 1,44 :K K are illustrated in Fig. 4.7 and


9

9

20

2021

21 15 8 8

22

22

77

140

13

15

27

26

26

33

271328

28

1

12 2

23

23

6

624

24

14

44

2525

55

16

1818

12

12 17

17

11

11

1010

19

19

16

Fig. 4.7: 1,2 1,24 :K K with EGL

12

2611

2527

27

25

12

26

11

28

28

9

910

10 29

29

8

8 3030

77

32

32

55

6

6

3

3

3434

4

4

33

33

3535

17

19

19

18 18

31

31

0

22

36

36

1

1

1723

23

14

14

13

13

2424

1515

2020

2222

16

16

21

21

Fig. 4.8: 1,2 1,44 :K K with EGL

45

Definition 4.2.10

A y-tree is a graph obtained from a path by appending an edge to a

vertex of a path adjacent to an end point and it is denoted by Yn where n is the

number of vertices in the tree.

Theorem 4.2.11

The y-tree Yn is edge – graceful for all n odd and n ≥ 11.

Proof

Let {v1, v2, ..., vn} be the vertices of Yn and {e1, e2, ..., en-1} be the edges

of Yn which are denoted as in Fig. 4.9.

v1

v2v3v4vn-1vn

en-1 e3e2

e1

. . .

. . .

Fig. 4.9: Yn with ordinary labeling


( )if e = i 1 ≤ i ≤ 1

2

n +

3

2

nf e +

= 5

2

n +

5

2

nf e +

= 3

2

n +

( )if e = i 7

2

n + ≤ i ≤ n – 1


( )if v+ = i 1 ≤ i ≤ 2

( )3f v+ = 6

46

( )if v+ = 2i – 1 4 ≤ i ≤ 1

2

n −

1

2

nf v++

= 0

( )if v+ = 3 2

2

i n+ −

3

2

n + ≤ i ≤

7

2

n +

( )if v+ = 2i – n – 1 9

2

n + ≤ i ≤ n

Clearly, the vertex labels are all distinct. Hence, the y-tree Yn is edge –

graceful for all n odd and n ≥ 11.

The EGL of Y9 and Y13 are illustrated in Fig. 4.10 and Fig. 4.11

respectively.

1

20348

6 57

1

7 26

4 385

Fig. 4.10: Y9 with EGL

1212

1110 8

105

84

9

37

06

115

9

4

73

6 22

1

1

Fig. 4.11: Y13 with EGL

Definition 4.2.12

A fire – cracker is a graph obtained from the concatenation of m copies

of the stars K1,n by linking one leaf from each. It is denoted by FC(1m, K1,n).

Theorem 4.2.13

The graph FC(1m, K1,n) is edge − graceful for all m, n even.

47

Proof

Let {w, w1, w2, ..., wm} ∪ {uij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} be the vertices and

{ fi : 1 ≤ i ≤ m} ∪ {eij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} be the edges of FC(1m, K1,n) which

are denoted as in Fig. 4.12.

f1

fm

w

f2

. . .

. . .

. . .

wm

um1um2

umn

em1em2

emn

. . .

. . .

. . .. .

.

w1

u1n

u11

u12

e11

e12

e1nw2

u21 u22

u2n

e21 e22

e2n

. . .

. . .

Fig. 4.12: FC(1m, K1,n) with ordinary labeling



q. There will be

2


With these pairs, we label the edges of FC(1m, K1,n) by the coordinates

of the pairs, in any order so that adjacent edges receive the coordinates of the

pairs. Then the induced vertex label are ( )f w+ = 0 ; ( )if w+ = ( )if f+ 1 ≤ i ≤ m

and the pendant vertices will have labels of the edges with which they are

incident and they are distinct. Hence, the graph FC(1m, K1,n) is edge − graceful

for all m, n even.

48

The EGL of FC(14, K1,4) and FC(14, K1,6) are illustrated in Fig. 4.13 and


12

11

0

9

11

516

516

12

7

14

7

14

9

1 20

191

2019

10

10

3

3

18 18

44

17

17

6

61515

8

8

13

13

2

2

Fig. 4.13: FC(14, K1,4) with EGL

16

15

0

13

15

228

20

722

20

16

17

19

11

19

11

1713

28 2

26

28 2

26

14

14

23

23

6

24624

55

25

25

4

4

3

327

271

112

1218

18

10

10

78

21

2199

Fig. 4.14: FC(14, K1,6) with EGL

49

Definition 4.2.14

The graph mGn is obtained from m copies of 1, 1,:n nK K by joining one

leaf of ith copy of 1, 1,:n nK K with the center of (i + 1)th copy of 1, 1,:n nK K

where 1 ≤ i ≤ m – 1.

Theorem 4.2.15

The graph mGn is edge – graceful for n is even and for all m.

Proof

Let {ui : 1 ≤ i ≤ m} ∪ {vi : 1 ≤ i ≤ m} ∪ {wi : 1 ≤ i ≤ m} ∪

{ uij, vij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} be the vertices and {eij, gij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} ∪

{ ai1, ai2 : 1 ≤ i ≤ m} be the edges of mGn which are denoted as in Fig. 4.15.

u1

u11 u12u1n

e11 e12e1n

a11

w1

a12

v1

g11g12 g1n

v11v12

w2=v1n

u2

u21 u22

u2n

e21 e22e2n

a22a21

v2

g21g22 g2n

v21v22

um

um1 um2 umn

em1 em2 emn

am1

wm

am2

vm

vm1

vm2 vmn

gm1gm2 gmn

......

......

..

.

...

......

...

...

......

...

...

...

Fig. 4.15: mGn with ordinary labeling

50


form (t, p – t), where 1 ≤ t ≤ 2

q. There are

2

q pair of solutions. With these

pairs, we label the edges by the coordinates of the pair in any order so that the

adjacent edges receive the coordinates of the pairs.


( )if u+ = ( )1if a 1 ≤ i ≤ m

( )if v+ = ( )2if a 1 ≤ i ≤ m

( )1f w+ = 0

( )if w+ = ( )( )1i nf g − 2 ≤ i ≤ m

And all the pendant vertices will have the labels of the edges with which

they are incident and are distinct.

Hence, the graph mGn is edge – graceful for n is even and for all m.

The EGL of 3G2 and 2G4 are illustrated in Fig. 4.16 and Fig. 4.17

respectively.

2

1

1 18

182

0

17

17

3

3

16

5

4

4 15

15

165 14

14

6

6

13

138

7

7 12

12

9

9

10

10

118 11

Fig. 4.16: 3G2 with EGL

51

1

1

3

20

20

2

2

19

19

3

0

18

18

4

4

17

17

5

5

16

168

6

615

15 7

7

14

14

8 13

13

11

11

10

1012

12

9

9

Fig.4.17: 2G4 with EGL

CHAPTER – V

Some More Strong Edge – Graceful Graphs

52

CHAPTER – V

SOME MORE STRONG EDGE – GRACEFUL

GRAPHS

5.1. Introduction

In the previous chapters, we have discussed edge – graceful labeling of

several families of trees. In this chapter, we present results on the strong

edge – gracefulness of some more graphs.

� Cm @ K1,n, m ≥ 3, n ≥ 2, m ≥ n when m, n are even.

� Cm @ K1,n, m ≥ 3, n ≥ 2, m ≥ n when m and n are odd.

� Cm @ K1,n, m ≥ 3, n ≥ 2, m > n when m is even and n is odd.

� M(Pn), for all n ≥ 4.

� Fl(n) (n ≥ 4), when n is even.

� Fl(n) (n ≥ 4), when n is odd.

� ( )3

tC for all t ≡ 1 (mod 6).

� ( )3


� ( )3


� ( )3


� ( )3


� ( )3


� Cn : Cn @ Wk (n ≥ 3, k ≥ 2), for all n and for k is even.

� Cm : Cn @ W2 (m, n ≥ 3), for all m > n where n and m are consecutive

integers.

� Cn : Cn @ W3 (n ≥ 3), for all n.

� Cm : Cn @ W3 (m, n ≥ 3), for all m, n and m > n where n and m are

consecutive integers.

� Cm : Cn @ W4 (n ≥ 3), for all m, n and m > n where n and m are


53

5.2. Cm @ K1,n graph

Definition 5.2.1

The graph Cm @ K1,n is obtained from the cycle Cm by attaching any one

leaf of the star K1,n to any one vertex of the cycle Cm.

Theorem 5.2.2

The graph Cm @ K1,n (m ≥ 3, n ≥ 2) is strong edge – graceful for all

m ≥ n and m, n are even.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of Cm @ K1,n and

{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '

ne } be the edges of Cm @ K1,n which are

denoted as in the Fig. 5.1.

v1

e1

v2

v3

e2

e3

vm

vm-1

em-1

em

'nv'

ne

'1e '

2e

'3e

'1v

'3v

'2v

'1nv −

'1ne − . .

.. .

.. . .

. . .

Fig. 5.1: Cm @ K1,n with ordinary labeling

We first label the edges of Cm @ K1,n as follows:

( )if e = i 1 ≤ i ≤ m

( )'if e = m + 2i 1 ≤ i ≤ n


( )if v+ = 2i – 1 1 ≤ i ≤ m

54

( )'if v+ = m + 2i 1 ≤ i ≤ n – 1

( )'nf v+ = n

Clearly, the vertex labels are all distinct. Hence, the graph Cm @ K1,n

(m ≥ 3, n ≥ 2) is strong edge – graceful for all m ≥ n and m, n are even.

The SEGL of C10 @ K1,10 and C12 @ K1,6 are illustrated in Fig. 5.2 and


1

3

57

9

11

13

15 17

19

1

23

4

5

6

7

89

10

30 10

12 14

16

18

20

22

24

2628

12 1416

18

20

22

24

2628

Fig. 5.2: C10 @ K1,10 with SEGL

11

32

53

7

49

511

6

13

7

158

179

19

1021

1123

12

24 6

14

16

18

20

22

14

16

18

20

22


55

Theorem 5.2.3

The graph Cm @ K1,n (m ≥ 3, n ≥ 2) is strong edge – graceful for all

m ≥ n when m and n are odd.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '


denoted as in the Fig. 5.1, of Theorem 5.2.2.


( )if e = i 1 ≤ i ≤ m

( )'if e = m – 1 + 2i 1 ≤ i ≤

– 1

2

n

( )'if e = m + 1 + 2i

1

2

n + ≤ i ≤ n - 1

( )'nf e = m + n


( )1f v+ = 2m + n + 1

( )if v+ = 2i – 1 2 ≤ i ≤ m

( )'if v+ =

11 2 1

21

1 2 12

nm i i

nm i i n

m n i n

− − + ≤ ≤

+ + + ≤ ≤ −

+ =


(m ≥ 3, n ≥ 2) is strong edge – graceful for all m ≥ n when m and n are odd.

56



34

3

57

9

11

13

15

17

19

1

23

4

5

6

7

8 9

10

22 22

12 16

18

20

24

26

2830

12 14 16

20

24

26

2830

21

11

18

32

14

32


32

1

3

25

3749

5

11

6

13

7

158

17 9 1910

21

11

20 20

1214

16

18

22

24

26

28

12 1416

18

22

2426

28


Theorem 5.2.4

The graph Cm @ K1,n (m ≥ 3, n ≥ 2) is strong edge – graceful for m > n

when m is even and n is odd.

57

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m

( )'if e = m + 2i 1 ≤ i ≤ n

Then the induced vertex labels are

( )if v+ = 2i – 1 1 ≤ i ≤ m

( )'if v+ = m + 2i 1 ≤ i ≤ n


(m ≥ 3, n ≥ 2) is strong edge – graceful for m > n when m is even and n is odd.



1

13

25

374

9

5

11

6

13

7

15 8 179

19

10

24 24

1214

16

18

20

22

22

20

18

16

1412


58

18 20

22

24

26

28

30

323436

383811

32

534

9

511 7

613

715

817

9

1910

21 11

2312

25

1327

14 29

15 31

16

18 2022

24

26

28

30

323436


5.3. Mirror Graph

Definition 5.3.1

Let G be a graph. Let G' be a copy of G. The mirror graph M(G) of G is

defined as the disjoint union of G and G' with additional edges joining each

vertex of G to its corresponding vertex in G'.

Theorem 5.3.2

The mirror graph M(Pn) is strong edge – graceful for all n ≥ 4.

Proof

Let {v1, v2, v3, ..., vn, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of M(Pn ) and

{ e1, e2, e3, ..., en-1, en+1, en+2, ..., e2n-1, e2n, e2n+1, ..., e3n-2} be the edges of M(Pn)

which are denoted as in the Fig. 5.8.

v1 v2 v3 vn-1 vn '1v '

2v '3v '

1nv −'nv

e3n-2

e2n+1

e2n

e2n-1

e2n-2en+1enen-1en-2e3e2e1

...

... ...

Fig. 5.8: M(Pn) with ordinary labeling

59

We first label the edges of M(Pn) as follows:

( )if e = i 1 ≤ i ≤ n – 1

( )if e = i + 1 n ≤ i ≤ 2n – 2

( )if e = n + 1 + i 2n – 1 ≤ i ≤ 3n – 2


( )if v+ = i – 1 1 ≤ i ≤ n – 1

( )nf v+ = 4n – 1 ; ( )'1f v+ = n

( )'if v+ = 2n – 1 + i 2 ≤ i ≤ n – 1

( )'nf v+ = n – 1

Clearly, the vertex labels are all distinct. Hence, the mirror graph M(Pn)

is strong edge – graceful for all n ≥ 4.

The SEGL of M(P6) and M(P9) are illustrated in Fig. 5.9 and Fig. 5.10

respectively.

0 1 2 3 4 23 6 13 14 15 16 5

1 2 3 4 5 7 8 9 10 11

23 22 21 20 19 18

Fig. 5.9: M(P6) with SEGL

0 1 2 3 4 5 6 7 35 9 19 20 21 22 23 24 25 8

1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17

35 34 33 32 31 30 29 28 27

Fig. 5.10: M(P9) with SEGL

60

5.4. Flower graph

Definition 5.4.1

A flower Fl(n) (n ≥ 3) is the graph obtained from a helm Hn by joining

each pendant vertex to the central vertex of the helm.

Theorem 5.4.2

The flower graph Fl(n) (n ≥ 4) is strong edge – graceful when n is even.

Proof

Let {v0, v1, v2, v3, ..., vn ,' ' '1 2 3, , v v v , ..., '

nv } be the vertices of Fl(n) and

{ e1, e2, e3, ..., e4n} be the edges of Fl(n) which are denoted as in the Fig. 5.11.

.

vn-1

vn-1'

e3n-1

e3n+2

e3

v0

en+4

v3' e2n+3v3

en+3e2

v2'

e2n+2

v2

e4n-1

en+2

e1

e4n

v1'

e2n+1

v1

en+1

en

e3n+1vn'e3nvn

e2n

en-1

. . .

. . .

. . .

Fig. 5.11: Fl(n) with ordinary labeling

We first label the edges of Fl(n) as follows:

( )if e = i 1 ≤ i ≤ 2n – 1

( )if e = i + 1 2n ≤ i ≤ 4n

61


( )if v+ = 2n + 1 + 2i 1 ≤ i ≤ n

( )'if v+ = 2i 1 ≤ i ≤ n – 2

( )'1nf v+

− = 2n – 1 ; ( )'nf v+ = n + 1

( )0f v+ = 0

Clearly, the vertex labels are all distinct. Hence, the flower graph Fl(n)

(n ≥ 4) is strong edge – graceful when n is even.

The SEGL of Fl(4) and Fl(8) are illustrated in Fig. 5.12 and Fig. 5.13

respectively.

15

12

77

2

1311416

15

1

11

10

2

17

1451317

4

0

3

9

56

Fig. 5.12: Fl(4) with SEGL

62

27

25

23

21

19

33

31

29

6

54

3

2

18

7

23

22

21

20

19

18

25

24

10

8

6

42

9

15

12

13

12

11

109

17

15

14

29

30

31

323326

27

28

0


Theorem 5.4.3

The flower graph Fl(n) (n ≥ 4) is strong edge – graceful when n is odd.

Proof

Let {v0, v1, v2, v3, ..., vn, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of Fl(n), and

{ e1, e2, e3, ..., e4n} be the edges of Fl(n) which are denoted as in the Fig. 5.11,

of Theorem 5.4.2.

We first label the edges of Fl(n) as follows:

( )if e = i 1 ≤ i ≤ 2n

( )if e = i + 1 2n + 1 ≤ i ≤ 4n


( )if v+ = 2n + 1 + 2i 1 ≤ i ≤ n

( )'if v+ = 2i 1 ≤ i ≤ n – 1

( )'nf v+ = n ; ( )0f v+ = 0

63

Clearly, the vertex labels are all distinct. Hence, the flower graph Fl(n)

(n ≥ 4) is strong edge – graceful when n is odd.

The SEGL of Fl(5) and Fl(9) are illustrated in Fig. 5.14 and Fig. 5.15

respectively.

19

17

15

13

21

4

15

3

14

213

1

12

5

16

8

6

4

2

5

21

20

19

1817

0

10

9

87

6


31

29

27

25

2321

37

35

336 5

4

3

2

19

8

7

1615

14

13

12

1110

18

17

24

23

22

2120

28

27

26

25

12

10

8

6

42

9

16

1432

33

34

353637

29

30

31

0


64

5.5. Friendship graph

Definition 5.5.1

The friendship graph ( )3

tC is the one-point union of t copies of the cycle C3.

Theorem 5.5.2


tC is strong edge – graceful for all t ≡ 1 (mod 6).

Proof

Let {v0, v1, v2, v3, ..., vt, ' ' '1 2 3, , v v v , ..., '

tv } be the vertices of ( )3

tC and

{ e1, e2, e3, ..., e3t} be the edges of ( )3

tC which are denoted as in the Fig. 5.16.

v2'

v1'

vt'

vt-1'

vt-1 v3

v2

v1

vt

e2t+2

e2t+1

e3t

e3t-1

e2t-3

e2t-2

e2t-1

e2t e1

e2

e3

e4

v0

. . .. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . . . . .

v3'

Fig. 5.16: ( )3

tC with ordinary labeling

We first label the edges of(t)3C as follows:

( )if e = i 1 ≤ i ≤ 3t

65


( )if v+ = 2t + 3i – 1 1 ≤ i ≤ 2 1

3

t +

2 1

3

ti

f v+++

= 3i – 2 1 ≤ i ≤ 1

3

t −

( )'if v+ = 2t + 3i 1 ≤ i ≤

2 1

3

t +

'2 1

3

ti

f v+++

= 3i – 1 1 ≤ i ≤ 1

3

t −

( )0f v+ = 2t +1

Clearly, the vertex labels are all distinct. Hence, the friendship graph

( )3


The SEGL of (7)3C and (13)

3C are illustrated in Fig. 5.17 and Fig. 5.18

respectively.

29 2826

25

23

22

20

1917

16

5

4

2

1

20

19

18

17

1615

21

11

12

10 98

7

6

5

432

1

14

13 15

Fig. 5.17: (7)3C with SEGL

66

34 495047

33

46

44

32

43

41

31

40

38

30

37

35

29

3432

283129

2728

11

39

10

8

38

7

5

37

4

2

36

1

53

35

52

12 3 4

5

6

7

8

9

10

11

12

13

14151617

18

19

20

21

22

23

24

25

26

27


Theorem 5.5.3



Proof

Let {v0, v1, v2, v3, ..., vt, ' ' '1 2 3, , v v v , ..., '


tC and


tC which are denoted as in the Fig. 5.16, of

Theorem 5.5.2.

67

We first label the edges of ( )3

tC as follows:

( )if e = i 1 ≤ i ≤ 3t


( )if v+ = 2t + 3i – 1 1 ≤ i ≤ 2

3

t

2

3

ti

f v+

+

= 3i – 3 1 ≤ i ≤ 3

t

( )'if v+ = 2t + 3i 1 ≤ i ≤

2

3

t

'2

3

ti

f v+

+

= 3i – 2 1 ≤ i ≤ 3

t

( )0f v+ = 2t + 1


( )3




respectively.

9 01

6 5

71

8

9

72

3

4

12

8

11

Fig. 5.19: (3)

3C with SEGL

68

2436

35

33

23

32

30

22

29

27

21

26

2420

2321

19

20

7

27

6

4

26

3

1

250

12 11

10

9

8

7

6

5

4321

18

17

16

15

1413

19


Theorem 5.5.4



Proof

Let {v0, v1, v2, v3, ..., vt, ' ' '1 2 3, , v v v , ..., '


tC and



Theorem 5.5.2.


tC as follows:

( )if e = i 1 ≤ i ≤ 3t


( )if v+ = 2t + 3i – 1 1 ≤ i ≤ 2( 1)

3

t +

2( 1)

3

ti

f v+++

= 3i – 1 1 ≤ i ≤ 2

3

t −

69

( )'if v+ = 2t + 3i 1 ≤ i ≤

2 1

3

t −

'2 1

3

ti

f v+−+

= 3i – 3 1 ≤ i ≤ 1

3

t +

( )0f v+ = 2t +1


( )3




respectively.

014

21

19

13

18

16

12

15

13

1211

3

2

15

11

87

6

5

4

3

21

10

9


70

3028242725

29 4243

40

28

39

37

27

36

34

26

33

31

252324

9

33

8

6

32

5

3

31

2

0

3045

14 13

12

11

10

9

8

7

65

4321

22

21

20

19

18

17

1615

23


Theorem 5.5.5



Proof

Let {v0, v1, v2, v3, ..., vt, ' ' '1 2 3, , v v v , ..., '


tC and



Theorem 5.5.2.


tC as follows:

( )if e = i 1 ≤ i ≤ 8

3

t

( )if e = i + 1 8

3

t + 1 ≤ i ≤ 3t

71


( )if v+ = 2t + 3i – 1 1 ≤ i ≤ 2

3

t

2

3

ti

f v+

+

= 3i – 2 1 ≤ i ≤ 3

t

( )'if v+ = 2t + 3i 1 ≤ i ≤

2

3

t

'2

3

ti

f v+

+

= 3i – 1 1 ≤ i ≤ 3

t

( )0f v+ = 0


( )3




respectively.

. . .

24 16 23

21

15

20

18

14

17

151314

5

19

4

2

18

1

8 76

5

4

321

12

11

10

9

0


72

45 443142

41

30

39

38

29

36

35

28

33

32

27

30

2926

27252611

3710

8

36

7

5

35

4

2

34

1

4832

47

14 1312

11

10

9

8

7

6

5

43

2124

23

22

21

20

19

18

17

1615

0


Theorem 5.5.6



Proof

Let {v0, v1, v2, v3, ..., vt, ' ' '1 2 3, , v v v , ..., '


tC and



Theorem 5.5.2.


tC as follows:

( )if e = i 1 ≤ i ≤ 8 1

3

t −

73

( )if e = i + 2 8 2

3

t + ≤ i ≤ 3t


( )if v+ = 2t + 3i – 1 1 ≤ i ≤ 2 1

3

t −

2 1

3

ti

f v+−+

= 3i – 2 1 ≤ i ≤ 1

3

t +

( )'if v+ = 2t + 3i 1 ≤ i ≤

2 1

3

t −

'2 1

3

ti

f v+−+

= 3i – 1 1 ≤ i ≤ 1

3

t +

( )0f v+ = 0


( )3




respectively.

3121 30 28

20

27

25

19

24

22

18

21

1917

188

267

5

25

4

2

24

1

10 9 8

7

6

5

4

3

211615

14

13

12

11

0

Fig. 5.25: (8)

3C with SEGL

74

52 36 5149

3548

46

34

45

43

33

4240

32

3937

31

36

3430

33312930

1444

13

11

43

10

8

42

7

5

41

42

40

1

5537

54

16 1514

13

12

11

10

9

8

7

6

5

43

212827

26

25

24

23

22

21

2019

1817

0


Theorem 5.5.7



Proof

Let {v0, v1, v2, v3, ..., vt, ' ' '1 2 3, , v v v , ..., '


tC and



Theorem 5.5.2.


tC as follows:

( )if e = i 1 ≤ i ≤ 3t

75


( )if v+ = 2t + 3i – 1 1 ≤ i ≤ 2 1

3

t +

2 1

3

ti

f v+++

= 3i – 2 1 ≤ i ≤ 1

3

t −

( )'if v+ = 2t + 3i 1 ≤ i ≤

2 1

3

t +

'2 1

3

ti

f v+++

= 3i – 1 1 ≤ i ≤ 1

3

t −

( )0f v+ = 0


( )3




respectively.

11 1617

1

12

2

10 119

13

10

14

6 5

4

3

21

7

8

0


76

38 372635

25

34

32

24

31

29

23

28

26

22

25

232122

8

30

7

5

29

4

2

28

1

41

2740

12 11 10

9

8

7

6

5

4

321

20

19

18

17

16

15

1413

0


5.6. Butterfly graph Cm : Cn @ Wk

Definition 5.6.1

Cm : Cn @ Wk denote two cycles of orders m and n sharing a common

vertex with k pendant edges attached at the common vertex.

Theorem 5.6.2

The Butterfly graph Cn : Cn @ Wk is strong edge – graceful for all n and

for k is even.

Proof

Let {v1, v2, ..., v2n-1, w1, w2, ..., wk} be the vertices of Cn : Cn @ Wk and

{ e1, e2, ..., e2n, '1e , '

2e , ..., 'ke } be the edges of Cn : Cn @ Wk which are denoted

as in Fig. 5.29.

77

vn-1

vn

v1v2

v2n-1 v2n-2

vn+2

vn+1

en

e1

e2

en-1

e2n

e2n-1

en+2

en+1

wk

w2w1

'ke

'2e

'1e

...

...

....

...

..

.

.

.

...

..

.

...

.

.

.

..

.

..

.

Fig. 5.29: Cn : Cn @ Wk with ordinary labeling


( )if e = i 1 ≤ i ≤ 2n

( )'1f e = 2n + 1

( )'if e = 2n + 2i – 2 2 ≤ i ≤ k


( )if v+ = 2i + 1 1 ≤ i ≤ n – 1

( )nf v+ = 2 2 5 2

1 2

p k k

k

− + > =

( )if v+ = 2i + 1 n + 1 ≤ i ≤ 2n – 1

( )1f w+ = 2n + 1

( )if w+ = 2n + 2i – 2 2 ≤ i ≤ k

Clearly, the vertex labels are all distinct. Hence, the Butterfly graph

Cn : Cn @ Wk is strong edge – graceful for all n and for k is even.

The SEGL of C4 : C4 @ W8 and C5 : C5 @ W6 are illustrated in the

Fig. 5.30 and Fig. 5.31 respectively.

78

5 2 3

1

19

47

3

11

5

6 13

7

158

99

1010

12

12

14

14

16

1618

1820

20

22 22

Fig. 5.30: C4 : C4 @ W8 with SEGL

5 2 3

1

23

54

13

6

7 15

17

1910

1111

12

12

14

14

16

1618

1820

20

7

9

3

9

8


Theorem 5.6.3

The Butterfly graph Cm : Cn @ W2 (m, n ≥ 3) is strong edge – graceful

for all m > n where n and m are consecutive integers.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv , w1, w2} be the vertices of

Cm : Cn @ W2 and {e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '

ne , f1, f2} be the edges of

Cm : Cn @ W2 which are denoted as in the Fig. 5.32.

79

w2 w1f2 f1

vn-1'

vn'

v1=v1'

en-2'

vm-1

em-1vm

v2'

v3'

e2'

e1'

e3'

v2

v3

e1

e2

e3e4

e5

em. .

.. .

.. .

.. .

.

. . .

. .

.

. . .. . .

. . .

en-1'

en'

Fig. 5.32: Cm : Cn @ W2 with ordinary labeling

We first label the edges of Cm : Cn @ W2 as follows:

( )if e = i 1 ≤ i ≤ m

( )'if e = m + i 1 ≤ i ≤ n

( )if f = m + n + i 1 ≤ i ≤ 2


( )if v+ = i + 1 i = 1

( )if v+ = 2i – 1 2 ≤ i ≤ m

( )'1f v+ = ( )1f v+

( )'if v+ = 2m – 1 + 2i 2 ≤ i ≤ n

( )if w+ = m + n + i 1 ≤ i ≤ 2


Cm : Cn @ W2 (m, n ≥ 3) is strong edge – graceful for all m > n where n and m

are consecutive integers.

The SEGL of C8 : C7 @ W2 and C9 : C8 @ W2 are illustrated in


80

1717

1513

11

9

75

8

76

5

4

32

2

16

16

29

27

25

23

2119

15

14

13

12

1110

9

3

1


1919

1715

13

11

9

7

9

87

6

5

43

2

18

18

33

31

29

27

2523

17

16

15

14

1312

11

5

23

1

21

10


Theorem 5.6.4

The Butterfly graph Cn : Cn @ W3 (n ≥ 3) is strong edge – graceful for

all n.

Proof

Let {v1, v2, v3, ..., vn, ' ' '1 2 3, , v v v , ..., '

nv , w1, w2, w3} be the vertices of

Cn : Cn @ W3 and {e1, e2, e3, ..., en, ' ' '1 2 3, , e e e, ..., '

ne , f1, f2, f3} be the edges of

Cn : Cn @ W3 which are denoted as in the Fig. 5.35.

81

w3

w2

w1

f3f2 f1

vn

vn-1

en-1

e3

v3

e2 v2

e1

en

v1

'ne

'nv

'1ne −

'1nv −

'3e

'2e

'3v

'2v

'1e

'1v

..

...

.

.

.

.

...

..

.

..

.

...

..

.

..

.

..

....

Fig. 5.35: Cn : Cn @ W3 with ordinary labeling

We first label the edges of Cn : Cn @ W3 as follows:

( )if e = i + 1 1 ≤ i ≤ n

( )'if e = n + 1 + i 1 ≤ i ≤ n

( )1f f = 1

( )1if f + = 2n + 1 + i 1 ≤ i ≤ 2


( )1f v+ = 4

( )if v+ = 2i + 1 2 ≤ i ≤ n

( )'1f v+ = ( )1f v+

( )'if v+ = 2n + 1 + 2i 2 ≤ i ≤ n

( )if w+ = ( )if f 1 ≤ i ≤ 3


Cn : Cn @ W3 (n ≥ 3) is strong edge – graceful for all n.

82

The SEGL of C8 : C8 @ W3 and C11 : C11 @ W3 are illustrated in


1919

17

15 98

7

6

5

4

1

1

33

31

29

27

2523

17

16

15

14

1312

113 5

2

21

10

13

11

94

7

18

18


2525 1

1

24

24

23

2119

17

15

13

119

7

5

4143

45

27

2931

33

35

37

39

4

12

1110

9

8

7

6

5 4

3

2

23

22 2120

19

18

17

1615

14

13


Theorem 5.6.5

The Butterfly graph Cm : Cn @ W3 (m, n ≥ 3) is strong edge – graceful

for all m > n where n and m are consecutive integers.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv , w1, w2, w3} be the vertices of


ne , f1, f2, f3} be the edges of

Cm : Cn @ W3 which are denoted as in the Fig. 5.38.

83

w3

w2

w1

f3f2 f1

vm

vm-1

em-1

e3

v3

e2 v2

e1

em

v1

'ne

'nv

'1ne −

'1nv −

'3e

'2e

'3v

'2v

'1e

'1v

..

...

.

.

.

.

...

..

.

..

.

...

..

.

..

.

..

....



( )if e = i + 1 1 ≤ i ≤ m

( )'if e = m + 1 + i 1 ≤ i ≤ n

( )1f f = 1

( )if f = 2n + 2i 2 ≤ i ≤ 3


( )1f v+ = 8

( )if v+ = 2i + 1 2 ≤ i ≤ m

( )'1f v+ = ( )1f v+

( )'if v+ = 2m + 1 + 2i 2 ≤ i ≤ n

( )if w+ = ( )if f 1 ≤ i ≤ 3


Cm : Cn @ W3 (m, n ≥ 3) is strong edge – graceful for all m > n where n and m

are consecutive integers.

84

The SEGL of C9 : C8 @ W3 and C10 : C9 @ W3 are illustrated in Fig. 5.39

and Fig. 5.40 respectively.

2222

1917

15

13

11

9

10

98

7

6

54

8

1

1

35

33

31

29

2725

18

17

16

15

1413

12

7

35

2

23

11

20

20


2424

21

15

9

7

11

10

8

7

6

43

8

1

1

37

35

33

31

2927

1918

17

16

1514

13

5

2 25

12

22

22

13

19

917

115

39

20


Theorem 5.6.6

The Butterfly graph Cm : Cn @ W4 (n ≥ 3) is strong edge – graceful for

all m > n where n and m are consecutive integers.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv , w1, w2, w3, w4} be the vertices of


ne , f1, f2, f3, f4} be the edges

of Cm : Cn @ W4 which are denoted as in the Fig. 5.41.

85

w3w2

w1f3f2f1

vm

vm-1

em-1

e3

v3

e2 v2

e1

em

v1

'ne

'nv

'1ne −

'1nv −

'3e

'2e

'3v

'2v

'1e

'1v

..

.

.

.

.

..

.

..

.

...

..

.

..

.

..

....

f4

w4

v6

v5

v4



( )if e = i 1 ≤ i ≤ m

( )'if e = m + i 1 ≤ i ≤ n

( )if f = 2n + 2i 1 ≤ i ≤ 4


( )if v+ = 2i – 1 1 ≤ i ≤ m

( )'1f v+ = ( )1f v+

( )'if v+ = 2m – 1 + 2i 2 ≤ i ≤ n

( )if w+ = ( )if f 1 ≤ i ≤ 4


Cm : Cn @ W4 (n ≥ 3) is strong edge – graceful for all m > n where n and m are


86

The SEGL of C9 : C8 @ W4 and C10 : C9 @ W4 are illustrated in Fig. 5.42


1824

1715

13

11

9

7

9

87

6

5

43

1

31

29

27

25

2321

17

15

14

13

1211

5

23

1

10

22

20 22

2024

18

33 16


2020

22

22

19

1715

13

11

9

75

3

3335

37

23

2527

29

31

10

98

7

6

5

4

3 2

1

19

18 17

16

15

14

1312

11

24

24

26

26

1


CHAPTER – VI

Strong Edge – Graceful Labeling of Some Simple Graphs

87

CHAPTER – VI

STRONG EDGE −−−− GRACEFUL LABELING OF SOME SIMPLE GRAPHS

6.1. Introduction

In the preceeding chapters we discussed edge − graceful labeling and

strong edge − graceful labeling of varieties of graphs.

In this chapter, we establish the strong edge – gracefulness of the

following graphs in different cases.

� Flag Fln, when n is even.

� Flag Fln, when n is odd.

� Comb Pn+ for all n.

� Mobius ladder Mn, when n is even.

� Mobius ladder Mn, when n ≡ 1 (mod 4).

� Mobius ladder Mn, when n ≡ 3 (mod 4).

� Wheel Wn, when n ≡ 0 (mod 4).




� Tortoise Tn, when n ≡ 1 (mod 6).






� Bn,m, n, m even, n ≠ m.

� Bn,m, n, m odd, n ≠ m.

� Bn,m, n, m even, n = m.

� Bn,m, n, m odd, n = m.

� Bn,m, m odd, n even, n < m.

� Bn,m, n odd, m even, n < m.

88

6.2. Flag graph Fln

Definition 6.2.1

The flag Fln is obtained by joining one vertex of Cn to an extra vertex

called the root.

Theorem 6.2.2

The flag graph Fln is strong edge-graceful when n is even.

Proof

Let {v1, v2, v3, ..., vn, vn+1} be the vertices of Fln and {e1, e2, ..., en, en+1}

be the edges of Fln which are denoted as in the Fig. 6.1.

v1

v2

v3

vn-1

vn

e1

e2e3

en-1

en

en+1

vn+1

..

.

..

. ..

.

.

.

.

..

.

..

.

..

.

..

.

Fig. 6.1: Fln with ordinary labeling

We first label the edges of Fln as follows:

( )if e = i 1 ≤ i ≤ n

( )1nf e + = n + 2

89


( )if v+ = 2i – 1 1 ≤ i ≤ n

( )1nf v++ = n + 2

Clearly, the vertex labels are all distinct. Hence, the flag graph Fln is

strong edge − graceful when n is even.

The SEGL of Fl12 is illustrated in the Fig. 6.2.

1

3

59

11

13

15

17

1921

23

12

1

2

3 45

6

7

8

910

11

14

14

7

Fig. 6.2: Fl12 with SEGL

Theorem 6.2.3

The flag graph Fln is strong edge − graceful when n is odd.

Proof

Let {v1, v2, v3, ..., vn, vn+1} be the vertices of Fln and {e1, e2, ..., en, en+1}

be the edges of Fln which are denoted as in the Fig. 6.1, of Theorem 6.2.2.

We first label the edges of Fln as follows:

90

( )if e = i 1 ≤ i ≤ n

( )1nf e + = n + 1


( )if v+ = 0 i = 1

( )if v+ = 2i – 1 2 ≤ i ≤ n

( )1nf v++ = n + 1

Clearly, the vertex labels are all distinct. Hence, the flag graph Fln is

strong edge − graceful when n is odd.

The SEGL of Fl11 is illustrated in the Fig. 6.3.

0

3

59

11

13

15

1719

21

11

1

2

3 45

6

7

8

910

12

12

7

Fig. 6.3: Fl11 with SEGL

6.3. Comb graph Pn �� K1

Definition 6.3.1

The comb Pn � K1 is the graph obtained from the path Pn by attaching

pendant edge at each vertex of the path. It is also denoted by Pn+.

91

Theorem 6.3.2

The comb graph Pn � K1 is strong edge − graceful for all n.

Proof

Let {v1, v2, v3, ..., vn, vn+1, ..., v2n} be the vertices of Pn � K1 and

{e1, e2, e3, ..., e2n−2, e2n−1} be the edges of Pn � K1 which are denoted as in the Fig. 6.4.

vn

e2n-1 e2n-2 en+2 en+1

v3 v2 v1

vn+1 vn+2 v2n

e1 e2 e3 e4

. . . . . . . . .

. . . . . . . . .

en

en-1

v2n-1

Fig. 6.4: Pn � K1 with ordinary labeling

We first label the edges of Pn � K1 as follows:

( )if e = i 1 ≤ i ≤ n

( )1nf e + = 2n

( )n if e + = 2n + (i – 1) 2 ≤ i ≤ n – 1


( )1f v+ = n

( )2if v++ = 2n + i 0 ≤ i ≤ 2n – 3

( )if v+ = i – 1 i = 2n

Clearly, the vertex labels are all distinct. Hence, the comb graph Pn � K1

is strong edge − graceful for all n.

92

The SEGL of P9 � K1, P12 � K1 are illustrated as in Fig. 6.5 and


26 27 28 29 30 31 32 33 17

1 2 3 4 5 6 7 8

25 24 23 22 21 20 19 18 9

25 24 22 21 20 19 18 923

Fig. 6.5: P9 �� K1 with SEGL

34 33 32 31 30 29 28 27 26 25

34 33 32 31 30 29 28 27 26 25

35 36 37 38 39 40 41 42 43 44

1 2 3 4 5 6 7 8 9

24 12

24 12

45 23

10 11

Fig. 6.6: P12 �� K1 with SEGL

6.4. Mobius ladder graph Mn

Definition 6.4.1

The Mobius ladder Mn is the graph obtained from the ladder Pn × P2 by

joining the opposite end points of the two copies of Pn.

Theorem 6.4.2

The Mobius ladder graph Mn (n ≥ 4) is strong edge − graceful when n is

even.

Proof

Let {v1, v2, v3, ..., vn, v1', v2', ..., vn'} be the vertices of Mn and

{ e1, e2, e3, ..., e5n−4} be the edges of Mn which are denoted as in the Fig. 6.7.

93

e3n-2e2n

e2n-3e2n-2

e2n-1

e4n-2 e5n-4

e1 e2 e3

e4n-3 e4n-4 e3n e3n-1v1 v3 vn-1 vn

'1v

'2v

'3v '

1nv −'nv. . . . . .

. . .

. . . . . .

v2

. . .

e4n-1

e2n+2 e2n+1

Fig. 6.7: Mn with ordinary labeling

We first label the edges of Mn as follows:

( )if e = i 1 ≤ i ≤ 2n – 2

( )if e = 2i – 2n + 3 2n – 1 ≤ i ≤ 3n – 2

( )if e = 2i – 4n + 4 3n – 1 ≤ i ≤ 4n – 3

( )if e = i + 2 4n – 2 ≤ i ≤ 5n – 4


( )if v+ = 4n – 2i 1 ≤ i ≤ n – 1

( )nf v+ = 2n + 1 : ( )'1f v+ = 1

( )'if v+ = 4i – 5 2 ≤ i ≤ n – 1

( )'nf v+ = n – 4

Clearly, the vertex labels are all distinct. Hence, the Mobius ladder

graph Mn (n ≥ 4) is strong edge − graceful when n is even.

The SEGL of M8 is illustrated in Fig. 6.8.

1 32 3 33 7 34 11 35 15 36 19 37 23 38 4

30 30 28 28 26 26 24 24 22 22 20 20 18 18 17

31 29 27 25 23 21 19 17

1 3 5 7 9 11 13

2 4 6 8 10 12 14

Fig. 6.8: M8 with SEGL

94

Theorem 6.4.3

The Mobius ladder graph Mn is strong edge − graceful for all n ≡ 1 (mod 4).

Proof

Let {v1, v2, v3,...,vn, v1' v2',...,vn'} be the vertices of Mn and {e1, e2, e3,...,e5n−4}

be the edges of Mn which are denoted as in the Fig. 6.7, of Theorem 6.4.2.


( )if e =

5 1 5 4

1 2 2

2 2 1

2 1 2

2 2 1 2 1 3 2

2 1 3 1

2 4 2 3 4 3

1 4 2 5 5

n i n

i i n

n i n

n i n

i n n i n

n i n

i n n i n

i n i n

− = − ≤ ≤ − = − − = − + + ≤ ≤ − + = −

− + ≤ ≤ − + − ≤ ≤ −


( )if v+ =

4 6 1

4 2 6 2 2

4 2 1 1

n i

n i i n

i n n i n

− = − − ≤ ≤ − − − − ≤ ≤

( )'if v+ =

4 3 1 2

4 9 3 1

4

n i i

i i n

n i n

− + ≤ ≤ − ≤ ≤ − − =


graph Mn is strong edge − graceful for all n ≡ 1 (mod 4).


34 35 35 36 3 37 7 38 11 39 15 40 19 41 23

30 32 26 30 24 28 22 26 20 24 18 22 16 20 13

33 31 29 27 25 23 21 17

1 3 5 7 9 11 13

2 4 6 8 10 12 14

44 5

18

16

15

19 17


95

Theorem 6.4.4

The Mobius ladder graph Mn is strong edge − graceful for all n ≡ 3 (mod 4).

Proof

Let {v1, v2, v3,...,vn, v1', v2',...,vn'} be the vertices of Mn and {e1, e2, e3,...,e5n−4}

be the edges of Mn which are denoted as in the Fig. 6.7, of Theorem 6.4.2.


( )if e =

1 2 2

2 1 2 1

2 1 2

2 2 1 2 1 3 2

2 4 2 3 1 4 3

1 4 2 5 5

5 4 5 4

i i n

n i n

n i n

i n n i n

i n n i n

i n i n

n i n

≤ ≤ − + = − − = − + + ≤ ≤ − − + − ≤ ≤ −

+ − ≤ ≤ − + = −


( )if v+ =

4 6 1

4 2 6 2 2

5 3 1 1

n i

n i i n

i n n i n

− = − − ≤ ≤ − − − − ≤ ≤

( )'if v+ =

4 3 1 2

4 9 3 2

4 8 1

2

n i i

i i n

n i n

n i n

− + ≤ ≤ − ≤ ≤ − − = − + =


graph Mn is strong edge − graceful for all n ≡ 3 (mod 4).


42 43 43 44 3 45 7 46 11 47 15 48 19 49 23

38 40 34 38 32 36 30 34 28 32 26 30 24 28 22

41 39 37 35 33 31 29 27

1 3 5 7 9 11 13

2 4 6 8 10 12 14

50 27

25

16

15

26 20

51 36 59 13

21 23

18 20

17 19

24 16 22 21


96

6.5. Wheel graph Wn

Definition 6.5.1

The wheel Wn is defined as Wn = Cn + K1, where Cn is the cycle of length n.

Theorem 6.5.2

The wheel graph Wn is strong edge − graceful for all n ≡ 0 (mod 4).

Proof

Let {v0, v1, v2, v3, ..., vn} be the vertices of Wn and {e1, e2, e3, ..., e2n} be

the edges of Wn which are denoted as in Fig. 6.11.

v3

e3

v2

e2

v1

e1vn

en

vn-1

en-1

vn-2en+2

en+1e2n

e2n-1

v 0

..

.

..

.

. . .

..

.

.

.

.

..

.

Fig. 6.11: Wn with ordinary labeling

We first label the edges of Wn as follows:

( )if e = i 1 ≤ i ≤ 2n


( )if v+ = i – 1 1 ≤ i ≤ n

( )0f v+ = n + 2

Clearly, the vertex labels are all distinct. Hence, the wheel graph Wn is

strong edge − graceful for all n ≡ 0 (mod 4).

97

The SEGL of W12 is illustrated in Fig. 6.12.

6 75

6

4

5

3

4

2

3

12

01

11

12

10

11

9

10

8

9

7

8

17 1819

20

21

222324

13

14

15

16

14

Fig. 6.12: W12 with SEGL

Theorem 6.5.3


Proof


the edges of Wn which are denoted as in Fig. 6.11, of Theorem 6.5.2.


( )if e = i 1 ≤ i ≤ n

( )if e = i + 1 n + 1 ≤ i ≤ 2n


( )if v+ = i 1 ≤ i ≤ n

( )0f v+ = n + 1



98


9 9 88

76

6

5

5

4

3

2

2

1181817

1716

16

15

15

14

14

28 29 3031323334

353620

21

13

13

12

1211

11

22

23

24

2526

3

1

10 10

27

7

37

419


Theorem 6.5.4


Proof




( )if e = i 1 ≤ i ≤ n

( )if e = i + 1 n + 1 ≤ i ≤ 2n


( )if v+ = i 1 ≤ i ≤ n

( )0f v+ = ( )31

2n +

99



The SEGL of W13 are illustrated in Fig. 6.14.

6 65

5

4

4

3

3

2

2

1

113

1312

12

11

11

10

10

9

9

88

21 22

23

24

25

262715

16

17

18

19

21

7 7

20


Theorem 6.5.5


Proof




( )if e = i 1 ≤ i ≤ 2n


( )if v+ = i – 1 1 ≤ i ≤ n

( )0f v+ = 3 5

2

n +

100




6 75

6

4

5

3

4

2

3

20

1415

14

13

11

12

10

11

9

10

8

9

23 2425

26

27

28

3016

18

19

20

21

25

7 8

22

1

29

113

12

17


6.6. Tortoise graph Tn

Definition 6.6.1

A tortoise Tn (n ≥ 4) is obtained from a path v1, v2,..., vn by attaching an

edge between vi and vn−i+ 1 for 1 ≤ i ≤ 2

n

.

Theorem 6.6.2

The tortoise graph Tn is strong edge − graceful for all n ≡ 1 (mod 6).

Proof

Let {v1, v2, v3, ..., vn} be n vertices of Tn and let 1 2 3 3( 1)

2

, , ,..., ne e e e −

be

the edges which are denoted as in Fig. 6.16.

101

v1 v2 v3

. . .

1

2

nv − 1

2

nv +

. . .

( )3 1

2

ne −

en+2

en+1

en

e1 e2 e3 en-1

3

2

nv + vn-2 vn-1 vn

en-21

2

ne − 1

2

ne +

. . .

Fig. 6.16: Tn with ordinary labeling

We first label the edges of Tn as follows:

( )if e = i 1 ≤ i ≤ 3( 1)

2

n −


( )if v+ = n – 2 + 3i 1 ≤ i ≤ 1

3

n −

( )if v+ = 3i – n – 2 2 1

3 2

n ni

+ −≤ ≤

1

2

nf v++

= n

( )if v+ = i – 1 3

12

ni n

+ ≤ ≤ −

( )nf v+ = 2n – 1

Clearly, the vertex labels are all distinct. Hence, the tortoise graph Tn is


102

The SEGL of T13 is illustrated in Fig. 6.17.

14

1

17

2

20

3

23

40

5

3

6

13

7

7

88

9

9

10

10

11

11

12

25

18

17

16

15

14

13

Fig. 6.17: T13 with SEGL

Theorem 6.6.3


Proof


2

, , ,..., ne e e e −

be

the edges which are denoted as in Fig. 6.16, of Theorem 6.6.2.


( )if e = i 1 ≤ i ≤ ( )3 1

2

n −


( )if v+ = n – 2 + 3i 1 ≤ i ≤ 3

n

( )if v+ = 3i – n – 2 3

3

n + ≤ i ≤

1

2

n −

1

2

nf v++

= n

103

( )if v+ = i – 1 3

2

n + ≤ i ≤ n − 1

( )nf v+ = 2n – 1




19

2

22

3

25

4

28

51

6

4

7

15

8

8

99

10

10

11

11

12

12

13

13

21

20

19

18

17

16

16

1 14

29

15


Theorem 6.6.4


Proof


2

, , ,..., ne e e e −

be

the edges which are denoted as in Fig. 6.16, of Theorem 6.6.2.


( )if e = i + 1 1 ≤ i ≤ ( )3 1

2

n −


( )1f v+ = n + 3

104

( )if v+ = n + 1 + 3i 2 ≤ i ≤ 2

3

n −

( )if v+ = 3i – n + 1 1 1

3 2

n ni

+ −≤ ≤

1

2

nf v++

= n + 2

( )if v+ = i + 2 3

12

ni n

+ ≤ ≤ −

( )nf v+ = 1




20 24 27 30 33 2 5 8 19 12 13 14 15 16 17 18 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

25

24

23

22

21

20

19

18


Theorem 6.6.5


Proof

Let {v1, v2, v3, ..., vn} be the vertices of Tn and 1 2 3 3 1

2

, , ,..., ne e e e−

be

the edges of Tn which are denoted as in the Fig. 6.20.

105

v1 v2 vn2

3

nv +1

2

nv+ 2

2

nv+ 1nv −

. . .

. . . . . .

. . .

en-1

31

2

ne−

en+1

en

. . .

en+2

1

3

nv −2

nv

e1

Fig. 6.20: Tn with ordinary labeling


( )if e = i 1 ≤ i ≤ 3

12

n −


( )if v+ = n – 2 + 3i 1 ≤ i ≤ 1

3

n −

( )if v+ = 3i – n – 2 2

3 2

n ni

+ ≤ ≤

( )if v+ = i – 1 1 12

ni n+ ≤ ≤ −

( )nf v+ = 2n – 1



106


20

2

23

3

26

4

29

50

6

3

7

6

9

9

1010

11

11

12

12

13

13

14

14

22

21

20

19

18

17

17

1 15

31

16

8

23

8


Theorem 6.6.6


Proof


2

, , ,..., ne e e e−

be

the edges of Tn which are denoted as in the Fig. 6.20, of Theorem 6.6.5.


( )if e = i 1 ≤ i ≤ 3

12

n −


( )if v+ = n – 2 + 3i 1 ≤ i ≤ 3

n

( )if v+ = 3i – n – 2 13 2

n ni+ ≤ ≤

( )if v+ = i – 1 1 12

ni n+ ≤ ≤ −

( )nf v+ = 2n – 1

107




13 16 19 22 1 4 6 7 8 9 10 23

1 2 3 4 5 6 7 8 9 10 11

17

16

15

14

13

12


Theorem 6.6.7


Proof


2

, , ,..., ne e e e−

be

the edges of Tn which are denoted as in the Fig. 6.20, of Theorem 6.6.5.


( )if e = i 1 ≤ i ≤ n – 2

( )if e = i + 1 n – 1 ≤ i ≤ 3

12

n −


( )if v+ = n – 1 + 3i 1 ≤ i ≤ 2

3

n −

( )if v+ = 3i – n – 1 1

3 2

n ni

+ ≤ ≤

108

( )if v+ = i 1 22

ni n+ ≤ ≤ −

( )1nf v+− = n

( )nf v+ = 1




19 22 25 0 3 6 8 9 10 11 12 14

2 3 4 5 6 7 8 9 10 11 12

21

20

19

18

17

16

1 14

1

16

15


6.7. Bn,m Graph

Definition 6.7.1

The graph Bn,m is defined as the graph obtained by joining the center u of

the star K1,n and the center v of another star K1,m to a new vertex w. It is also

denoted by 1, 1,:n mK K .

We now discuss the strong edge – graceful labeling of Bn,m in different

cases.

109

Theorem 6.7.2

The graph Bn,m is strong edge – graceful for all m, n even and m ≠ n.

Proof

Let {u, v, w, v1, v2, ..., vm, u1, u2, ..., un} be the vertices and

{ ai : 1 ≤ i ≤ n} ∪ {bi : 1 ≤ i ≤ m} ∪ {e, e'} be the edges of Bn,m which are

denoted as in Fig. 6.24.

u vwe e'

a1a2

. . .

. . .

an

u1

u2

un

v1

v2

vm

. . .

. . .

bm

b1b2

Fig. 6.24: Bn,m with ordinary labeling

Consider the Diophantine equation x1 + x2 = 2p. The solutions are of the

form (t, 2p – t) where p + 1 ≤ t ≤ 3

2

q

. There will be 3

2

qp

− pair of

solutions.

With these pairs, we label the edges of K1,n and K1,m by the coordinates

of the pair in any order so that adjacent edges receive the coordinates of the

pairs.

Also, we label the edges e and e' as follows:

( )f e = 1 ; ( )'f e = 4


( )f u+ = 1 ; ( )f v+ = 4 ; ( )f w+ = 5

110

And all other pendant vertices will have labels of the edges with which

they are incident and they are all distinct. Hence, the graph Bn,m is strong edge –

graceful for all m, n even and m ≠ n.

The SEGL of B6,8 is illustrated in Fig. 6.25.

11

54

4

22

12

23

11

24

10

10

24

11

23

1222

1816

19

15

20

14

2113

18 1619

15

20

14

2113

Fig. 6.25: B6,8 with SEGL

Theorem 6.7.3

The graph Bn,m is strong edge – graceful for all m, n odd and m ≠ n.

Proof



denoted as in Fig. 6.24, of Theorem 6.7.2.


form (t, 2p – t) where p + 1 ≤ t ≤ 3

2

q

. There will be 3

2

qp

− pair of

solutions.

Among 3

2

qp

− solutions, choose the pair

3,

2

qq

. We now label

the edges e and e' as follows:

( )'f e = q ; ( )1f a = n + 1 ; ( )1f b = 1 ; ( )f e = n + m + 4

111

As we have labeled e and e', there will be even number of edges left out

at each of u and v to be labeled.

We label these edges as was done in Theorem 6.7.2.


( )f u+ = 2n + m + 5 ; ( )f v+ = n + m + 3 ; ( )f w+ = 0

And, all other pendant vertices will have the respective labels of the

edges with which they are incident and are distinct. Hence, the graph Bn,m is

strong edge – graceful for all m, n odd and m ≠ n.


2820

0

1819

8

8

25

13

26

12

27

11

1 14

24

15

23

16

22

1721

2117

22

16

23

152414125

13

26

12

27

11


Theorem 6.7.4

The graph Bn,m is strong edge – graceful for all m, n even and m = n.

Proof




112


form (t, 2p – t) where p + 1 ≤ t ≤ 3

2

q

. There will be 3

2

qp

− pair of

solutions.



pairs.


( )f e = 1 ; ( )'f e = n + 1


( )f u+ = 1 ; ( )f v+ = n + 1 ; ( )f w+ = n + 2

And the pendant vertices will have labels of the edges with which they

are incident and they are all distinct. Hence, the graph Bn,m is strong edge –

graceful for all m, n even and m = n.


11

8

77

9

21

10

20

11

19

12

18

13

17

14

16

16

14

17

13

18129

21

10

20

11

19


113

Theorem 6.7.5

The graph Bn,m is strong edge – graceful for all m, n odd and m = n.

Proof





form (t, 2p – t) where p + 1 ≤ t ≤ 3

2

q

. There will be 3

2

qp

− pair of

solutions.

Among 3

2

qp

− solutions, choose the pair

3,

2

qq

. We now label

the edges e and e' as follows:

( )'f e = q ; ( )1f a = n ; ( )1f b = 1 ; ( )f e = 2n + 4


( )f u+ = 3n + 4 ; ( )f v+ = q + 1 ; ( )f w+ = 0

And, all other pendant vertices will have the respective labels of the

edges with which they are incident and are distinct. Hence, the graph Bn,m is

strong edge – graceful for all m, n odd and m = n.


1310

0

89

3

7

11

1

6

12

12

6

13

7

11


114

Theorem 6.7.6

The graph Bn,m is strong edge – graceful for all n even, m odd and n < m.

Proof





form (t, 2p – t) where p + 1 ≤ t ≤ 3

2

q

. There will be 3

2

qp

− pair of

solutions.



pairs.


( )f e = 1 ; ( )'f e = 1

2

q + ; ( )1f b = q + 1


( )f u+ = 1 ; ( )f v+ = ( )3 1

2

q +; ( )f w+ =

3

2

q +

And the pendant vertices will have labels of the edges with which they

are incident and they are all distinct. Hence, the graph Bn,m is strong edge –

graceful for all n even, m odd and n < m.

115


11

109

27

11

25

12

24

13

23

23

13

24

12

2511

1422

15

21

16

20

1719

14 2215

21

16

20

1719

18

18


Theorem 6.7.7

The graph Bn,m is strong edge – graceful for all n odd, m even and n < m.

Proof

Proof can be obtained from the proof of Theorem 6.7.6 by interchanging

n and m.

CHAPTER – VII

Strong Edge – Graceful Labeling of Some Cycle Related Graphs

116

CHAPTER – VII


OF SOME CYCLE RELATED GRAPHS

7.1. Introduction

In the preceeding chapters, we have established the strong edge − graceful

labeling of several families of graphs. In continuation with this, we study the

following graphs and verify their strong edge − graceful property through the

results given below.

� Cn @ Pt (n ≥ 3, t ≥ 1) for n is even and t is odd.

� Cn @ Pt (n ≥ 3, t ≥ 1) for n is even and t is even except t = n + 2.

� Cn @ Pt (n ≥ 3, t ≥ 1) for n is even and t = n + 2.

� Cn @ Pt (n ≥ 3, t ≥ 1) for n is odd and t is even.

� Cn @ Pt (n ≥ 3, t ≥ 1) for n is odd and t is odd except t = n + 2.

� Cn @ Pt (n ≥ 3, t ≥ 1) for n is odd and t = n + 2.

� K2 � Cn ( n ≥ 3) for all n.

� G(n) (n ≥ 4) for all n ≡ 0, 2 (mod 4).

� G(n) (n ≥ 4) for all n ≡ 1 (mod 4).

� G(n) (n ≥ 4) for all n ≡ 3 (mod 4).

� SF(n) (n ≥ 3) for all n ≡ 0 (mod 4).

� SF(n) (n ≥ 3) for all n ≡ 2 (mod 4).

� SF(n) (n ≥ 3) for all n ≡ 1 (mod 4).

� SF(n) (n ≥ 3) for all n ≡ 3 (mod 4).

� Sn for n is even.

� Sn for n is odd.

117

7.2. Kite graph Cn @ Pt

Definition 7.2.1

The graph Cn @ Pt (n ≥ 3, t ≥ 1) is obtained from the cycle Cn by

attaching a path of length t to any one vertex of Cn.

Theorem 7.2.2

The graph Cn @ Pt is strong edge – graceful when n is even and t is odd.

Proof

Let {v1, v2, ... vn+t} be the vertices of Cn @ Pt and {e1, e2, …, en, en+1,

en+2, ... en+t} be the edges of Cn @ Pt as shown in the Fig.7.1.

en+t

vn+t-1 vn+tvn+2vn+1

en+1 en+2

e1

e2e3

en-1

en

v1

v2v3

vn

v4

. . .vn-1

. . .

Fig. 7.1: Cn @ Pt with ordinary labeling

We first label the edges of Cn @ Pt as follows:

( )if e = (n + t + 1) – i 1 ≤ i ≤ n + t


( )1f v+ = 3 1 if 3 1 2

1 if 3 1 2

n t n t p

t n n t p

+ + + + < − + + + ≥

( )if v+ = 2(n + t) + 3 – 2i 2 ≤ i ≤ n

( )if v+ = 2(n + t) + 1 – 2i n + 1 ≤ i ≤ n + t

Clearly, the vertex labels are all distinct. Hence, the graph Cn @ Pt is

strong edge – graceful when n is even and t is odd.

118

The SEGL of C2 @ P3, C4 @ P7 and C6 @ P5 are illustrated in the

Fig. 7.2, Fig. 7.3 and Fig. 7.4 respectively.

35

3 29 2

1

1

5

4

Fig. 7.2: C2 @ P3 with SEGL

3

23

519 4

1

1

21

17

1110

9 8

7

4

9

5

11

6

13

7


1

1

19

0

21

7

11

10

6

13

3

2

5

3

7

4

9

5

9

17

8

15


119

Theorem 7.2.3

The graph Cn @ Pt is strong edge – graceful when n is even and t is even

except for t = n + 2.

Proof

Let {e1, e2, ..., en, en+1, ..., en+t} be the edges of Cn @ Pt as defined in the

Fig. 7.1, of Theorem 7.2.2.

We label the edges of Cn @ Pt as follows:

( )if e = i 1 ≤ i ≤ n + t


( )1f v+ = 2(n + 1)

( )if v+ = 2i – 1 2 ≤ i ≤ n

( )if v+ = 2i + 1 n + 1 ≤ i ≤ n + t – 1

( )n tf v++ = n + t


strong edge – graceful when n is even and t is even except for t = n + 2.



64

7

3

8

85 6 7

15131193

1

2 Fig. 7.5: C2 @ P6 with SEGL

120

106

11

5

21

107 8 9

191715135

3

7

12

43

12

1211

23


148

15

7

12

129 10 11

23211917

5 3

7

1

2

4

3

95 11

6


Theorem 7.2.4

The graph Cn @ Pt is strong edge − graceful when n is even and t = n + 2.

Proof



We label the edges of Cn @ Pt as follows:

( )if e = i 1 ≤ i ≤ n + t – 1

( )if e = i + 2 i = n + t

121


( )if v+ =

1 1

2 1 2

2 1 1 2

1

2

n t i i

i i n

i t n n i n t

n t i i n t

i i n t

+ − + = − ≤ ≤ + − − + ≤ ≤ + − + − = + −

+ = +


strong edge − graceful when n is even and t = n + 2.



67

3

9

1

4

2

3 5 8

1 8


1011

5

13

3

6

7

5 7 8

15 17

12

3 4

9 12

1 12


1415

7

17

3

8

11

5

9 10

19 21

1

2

9

6

11 12

23 25

3

7

4

5

13 16

1 16


122

Theorem 7.2.5

The graph Cn @ Pt is strong edge – graceful when n is odd and t is even.

Proof



We label the edges as follows:

( )if e = (t + n + 1) – i 1 ≤ i ≤ n + t


( )1f v+ = 3 1 if 3 1 2

1 if 3 1 2

n t n t p

t n n t p

+ + + + < − + + + ≥

( )if v+ = 2(n + t) + 3 – 2i 2 ≤ i ≤ n

( )if v+ = 2(n + t) + 1 – 2i n + 1 ≤ i ≤ n + t


strong edge – graceful when n is odd and t is even.

The SEGL of C3 @ P4 and C5 @ P4 are illustrated in the Fig. 7.11


2

13

3 1

11

7

5

657

1234


123

0

17

3 1

11

9

5

757

1234

8

613

15


Theorem 7.2.6

The graph Cn @ Pt is strong edge – graceful when n is odd and t is odd

except when t = n + 2.

Proof




( )if e = i 1 ≤ i ≤ n + t


( )1f v+ = 2(n + 1)

( )if v+ = 2i – 1 2 ≤ i ≤ n

( )if v+ = 2i + 1 n + 1 ≤ i ≤ n + t – 1

( )n tf v++ = n + t


strong edge – graceful when n is odd and t is odd except when t = n + 2.

The SEGL of C5 @ P9 and C5 @ P5 are illustrated in the Fig. 7.13 and


124

5

4

3

9

76

5

12

7

1

23

11

2

314

148 9 10 12 13

27252119171513


5

4

3

9

76

5

12

7

12

38 9 10

1019171513


Theorem 7.2.7

The graph Cn @ Pt is strong edge – graceful when n is odd and t = n + 2.

Proof




( )if e = i 1 ≤ i ≤ n + t – 1

( )n tf e + = n + t + 2


( )1f v+ = n + t

( )if v+ = 2i – 1 2 ≤ i ≤ n

( )if v+ = 2i + t – n – 1 n + 1 ≤ i ≤ n + t – 2

125

( )if v+ = 1 i = n + t – 1

( )if v+ = i + 2 i = n + t


strong edge – graceful when n is odd and t = n + 2.

The SEGL of C3 @ P5 and C5 @ P7 are illustrated in the Fig. 7.15 and


3

54

5

8

1

7 10

101131192

3

6


3

76

4

12

1

9 10

21191715133

9

8 11 14

141

5

2

7 5


7.3. K2 �� Cn graph

Definition 7.3.1

The graph K2 � Cn is obtained by attaching the cycle Cn at each end

point of K2.

126

Theorem 7.3.2

The graph K2 � Cn (n ≥ 3) is strong edge – graceful for all n.

Proof

Let {v1, v2, ..., vn, '1v , '

2v , …, 'nv } be the vertices of K2 � Cn and

{ e1, e2, ... en+1, '1e , '

2e , …, 'ne } be the edges of K2 � Cn as shown in Fig. 7.17.

vn'

en+1

vn-1v5

v4

v3

v2

v1

vn

e1

e2e3

e4

e5en

v1'

v2'

v3'

vn-1'

vn-2'

e1'

e2'

e3'

en-1'

en'

. . .

. . .

Fig. 7.17: K2 �� Cn with ordinary labeling

We first label the edges of K2 � Cn as follows:

( )if e = i 1 ≤ i ≤ n + 1

'( )if e = n + 1 + i 1 ≤ i ≤ n


( )if v+ = 2i + 1 1 ≤ i ≤ n – 1

( )nf v+ = 2n + 2

( )'if v+ = 2n + 3 + 2i 1 ≤ i ≤ n − 2

( )1 'nf v+− = 1

( )'nf v+ = 4

127

Clearly, the vertex labels are all distinct. Hence, the graph K2 � C

(n ≥ 3) is strong edge – graceful for all n.

The SEGL of K2 � C6 and K2 � C9 are illustrated in the Fig. 7.18


47

6

14

5 119

7

1

5 3

17 19

21

231

2

3

4 8

9

10

11

12

13

Fig. 7.18: K2 �� C6 with SEGL

410

9

20

158

7

6

1

5

4

17

25

29

31

33

9

11

13

11

2314

15

16

177

3 52

3

1

12 1327

3518

19

Fig. 7.19: K2 �� C9 with SEGL

7.4. Gear graph G(n)

Definition 7.4.1

The gear graph is obtained from the wheel Wn by subdividing all the

edges of its cycle and it is denoted by G(n).

128

Theorem 7.4.2

The gear graph G(n) (n ≥ 4) is strong edge − graceful for all n ≡ 0, 2 (mod 4).

Proof

Let { v0, v1, v2, v3, ..., vn, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of G(n) and

{ e1, e2, e3, ..., en, ' ' '1 2 3, , e e e, ..., '

ne , '1f , '

2f , '3f ,..., '

nf } be the edges of G(n) as

denoted as in Fig. 7.20.

. . . . . .

e6'e6

v6'

e5'v6

v5'e5

v5e4'

v4'e4

v4e3'v3'e3

v3e2'v2'

e2

v2

e1'

v1'e1

v1

en'vn'

envn

en-1'vn-1'

en-1vn-1

fn-2'fn-1

' fn' f1

'f2

'

. . .

. . .

. . .

. . . . . .

. . . . .

.. . .

. . .

. . .

. . . . . .. . .

. . .. . .. .

.

vo

Fig. 7.20: G(n) with ordinary labeling

We first label the edges of G(n) as follows:

( )if e = 3

12

ni+ + 1 ≤ i ≤ n

( )'if e = i 1 ≤ i ≤ n

( )'if f = 3 2 1

2

2 1 12

nn i i

nn i i n

+ − ≤ ≤ + − + ≤ ≤

129


( )0f v+ = 0

( )if v+ =

7 61

24

1 2 12 2

22 2

ni

n ni i

n ni i n

+ =

+ + − ≤ ≤ + − + ≤ ≤

( )'if v+ =

3 22 1

2

ni i n

+ + ≤ ≤

Clearly, the vertex labels are all distinct. Hence, the gear graph G(n)

(n ≥ 4) is strong edge − graceful for all n ≡ 0, 2 (mod 4).

The SEGL of G(8), G(10), G(12) and G(14) are illustrated in Fig. 7.21,

Fig.7.22, Fig. 7.23 and Fig. 7.24 respectively.

10 4 2117

9

3

19

16

8

2

17

15

71

15143182921

4

7

27

20

3

6

25

19

25

23 18

12 22

23

24

25910

11 0

Fig. 7.21: G(8) with SEGL

130

2

628

22 12

526

21

114

2420

103

2219

92

2018

8

118

1738

1036

26

59

3425

4 8

32 24

37

3023

14 15

27

28

29

3031

11

12

13

0


2

733

2614

6 31

25 13

529

24

12

4

2723

113

2522

102

2321

9

12045

1243

316

1141

30

510

3929

49

37 28

38

3527

1832

33

34

353637

13

14

15

16

17

0

21


131

2

838

3016

736

29 156 3428 14

532

2713

4

3026

123

2825

112

2624

10

124

2352

1450

367

1348356

1246

345

1144

33

410

42 32

39

4031

20 21 3738

39

40

41

42431516

17

18

19

0


Theorem 7.4.3

The gear graph G(n) is strong edge − graceful for all n ≡ 1 (mod 4).

Proof

Let{v0, v1, v2, v3, ..., vn, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., en, ' ' '1 2 3, , e e e, ..., '

ne , '1f , '

2f , '3f ,..., '


denoted in Fig. 7.20, of Theorem 7.4.2.

We first label the edges of G(n) as follows:

( )'if e = i 1 ≤ i ≤ n

( )if e = 3 1

12

ni

− + + 1 ≤ i ≤ n

( )'if f =

13 1 1

21

2 1 2

nn i i

nn i i n

− + − ≤ ≤ + + − ≤ ≤

132


( )0f v+ = 0

( )if v+ =

7 51

21 1

22 2

1 3

2 2

ni

n ni i

n ni i n

+ =

− + + ≤ ≤ + + − ≤ ≤

( )'if v+ =

3 12

2

ni

+ + 1 ≤ i ≤ n

Clearly, the vertex labels are all distinct. Hence, the gear graph G(n) is


The SEGL of G(9) and G(13) are illustrated in Fig. 7.25 and Fig. 7.26

respectively.

1

524

19 9

422

18

8

3

2017

72

1816

6

116

1534

932

234

830

223

7

2821

2 6

2620

14

24

25

262710

11

12

13

0


133

1

734

2713

6 32

2612

530

2511

428

24

103

2623

92

2422

8

122

2148

1346

336

1244325

1142

314

10

4030

39

38 29

28

3628

19 2034

35

36

37

38391415

16

17

18

0


Theorem 7.4.4

The gear graph G(n) is strong edge − graceful for all n ≡ 3 (mod 4).

Proof

Let{v0, v1, v2, v3, ..., vn, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., en, ' ' '1 2 3, , e e e, ..., '

ne , '1f , '

2f , '3f ,..., '


denoted in Fig. 7.20, of Theorem 7.4.2.


( )if e = i 1 ≤ i ≤ n

( )'if e =

3 11

2

ni

− + + 1 ≤ i ≤ n

( )'if f =

13 1 1

21

2 1 2

nn i i

nn i i n

− + − ≤ ≤ + + − ≤ ≤

134


( )0f v+ = n + 1

( )if v+ =

1 11

2 21 1

12 2

n ni i

n ni i n

+ − + ≤ ≤ − + − ≤ ≤ −

( )nf v+ = 7 5

2

n +

( )'if v+ =

3 12

2

ni

+ + 1 ≤ i ≤ n

Clearly, the vertex labels are all distinct. Hence, the gear graph G(n) is


The SEGL of G(11) and G(15) are illustrated in Fig.7.27 and Fig. 7.28

respectively.

1

629

2311

5 27

2210

425

219

323

20

82

2119

71

1918

41

1139

281037

274

935

263

8

3325

2 7

3124

17 29

30

31

32331213

14

15

16

12

5


135

1

839

3115

7 37

30 146 3529 13

533

2812

431

27113

2926

102

2725

91

2524

55

1553

387

1451

376

1349

365

1247

354

1145

34

3 10

43 33

29

4132

22 2339

40

41

42

4344

451617

18

19

20

21

16


7.5. Sunflower graph SF(n)

Definition 7.5.1

A sunflower graph SF(n) is defined as a graph obtained by starting with

an n-cycle with consecutive vertices v1, v2, v3, ..., vn and creating new vertices

w1, w2, w3, ..., wn with wi connected to vi and vi+1.

Theorem 7.5.2

The sunflower graph SF(n) (n ≥ 3) is strong edge – graceful for all

n ≡ 0 (mod 4).

Proof

Let {v1, v2, v3, ..., vn, w1, w2, w3, ..., wn} be the vertices of SF(n) and

{ e1, e2, e3, ..., e3n} be the edges of SF(n) are denoted as in the Fig. 7.29. It

consists of 2n vertices and 3n edges.

136

e2n+3

e4

v1

wn-1

wn w1

w2

w3

e3n-1

e3n e2n+1

e2n+2

v2

e3

e2e1e2n

e2n-1

. . . . . .

. . . . . .

. . .

. . .

. . . . . .

v3

v4

vn

vn-1

e5

e6

e2n-2

e2n-3

Fig. 7.29: SF(n) with ordinary labeling

We first label the edges of SF(n) as follows:

( )if e = i 1 ≤ i ≤ 2n + 1

( )if e = 5n + 2 – i 2n + 2 ≤ i ≤ 3n


( )1f v+ = 2n + 4 ; ( )2f v+ = n + 6

( )2if v++ = 2n + 6 + 2i 1 ≤ i ≤ n – 4

( )1nf v+− = 0 ; ( )nf v+ = 2

( )if w+ = 4i – 1 1 ≤ i ≤ n

Clearly, the vertex labels are all distinct. Hence, the sunflower graph

SF(n) (n ≥ 3) is strong edge – graceful for all n ≡ 0 (mod 4).

The SEGL of SF(8) and SF(12) are illustrated in Fig. 7.30 and Fig. 7.31

respectively.

137

0

30

26

6

2

31

3 7

11

15

18

17 14

2420

5

432

1

1923

27

14

13

12

11 10

9

8

7

16

15

28

24

23

22

2120

19

Fig. 7.30: SF(8) with SEGL

27

23

19

15

11

73

47

43

39

35

31

1516

17

18

19

20

21

22

23

24

12 3

4

5

6

7

8

9

10

11

12

1314

44

42

40

38

36

34

32

18

28

2

0

46

29

30 31

32

33

34

35

3625

26

27

28


138

Theorem 7.5.3

The sunflower graph SF(n) is strong edge − graceful for all n ≡ 2 (mod 4).

Proof


{ e1, e2, e3, ..., e3n} be the edges of SF(n) which are denoted as in the Fig. 7.29,

of Theorem 7.5.2. It consists of 2n vertices and 3n edges.


( )if e = i 1 ≤ i ≤ 2n ; ( )2 1nf e + = 5n + 2 – i

( )if e = 5n + 1 – i 2n + 2 ≤ i ≤ 3n


( )1f v+ = 3n + 3 ; ( )2f v+ = 2n + 5

( )2if v++ = 2n + 4 + 2i 1 ≤ i ≤ n – 3

( )nf v+ = 0 ; ( )if w+ = 4i – 1 1 ≤ i ≤ n


SF(n) is strong edge − graceful for all n ≡ 2 (mod 4).


respectively.

36

38

0

33

25

26

28

2235

39

3 7

28

29

21

31

2

1

3

4

19

20

27 23

19

15

11

31

12

11

10

9

8

7

6

5

18

17

16

15

14

13

30

32

34

27

26

2524

23


139

7

11

15

19

23

27

3135

39

43

47

51

55

3

1817 16 15

14

13

12

11

10

9

8

7

6

54

321

28

27

26

25

24

23

22

21

20

1946

44

42

40

38

36

3433

45

0

54

52

50

48

34 35

36

37

38

39

40

4143

29

30

31

32

33


Theorem 7.5.4

The sunflower graph SF(n) is strong edge – graceful for all n ≡ 1 (mod 4).

Proof





( )if e = i 1 ≤ i ≤ 2n

( )if e = 5n + 1 – i 2n + 1 ≤ i ≤ 3n


( )1f v+ = 3n + 2

( )if v+ = 2n + 2i 2 ≤ i ≤ n – 1

( )nf v+ = 0

( )if w+ = 4i – 1 1 ≤ i ≤ n

140


SF(n) is strong edge – graceful for all n ≡ 1 (mod 4).


respectively.

21

20

19

27

26

26

24

3

4

1 2

3

7

1131

35

17

18

29 22

16 0

27

23 19

158

7

6

5

91011

12

13

14

1525

24

28

233022

32

34


31 27

23

19

15

11

73

51

47

43

39

35 1615 14

13

12

11

10

9

8

7

6

54

32126

25

24

23

22

21

20

19

18

17

4042

44

46

48

50

0

4130

32

34

36

38

34

3332

31

30

29

28

27

3938

37

36

35


141

Theorem 7.5.5

The sunflower graph SF(n) is strong edge − graceful for all n ≡ 3 (mod 4).

Proof





( )if e = i 1 ≤ i ≤ 2n

( )if e = 5n + 2 – i 2n + 1 ≤ i ≤ 3n


( )1f v+ = 3n + 4

( )1if v++ = 2n + 4 + 2i 1 ≤ i ≤ n – 3

( )1nf v+− = 0 ; ( )nf v+ = 2

( )if w+ = 4i – 1 1 ≤ i ≤ n


SF(n) is strong edge − graceful for all n ≡ 3 (mod 4).


respectively.

142

2723

19

15

11

73

43

39

35

31 14 13 1211

10

9

1 2 3

4

5

6

7

8

22

21

20

19

18

17

16

15

40 38

36

34

32

30

2837

2

0

42

27

28 29

30

31

32

3334

24

25

26


35

1718 16 15

31

2714

1312

11

23

1019

15

11

7

359

55

51

47

43

39

98

76

54

32130

2928

27

26

25

24

23

22

21

2019

3738

40

39

41

42

43

44

454632

33

34

35

36

5048

46

44

42

40

38

36492

0

58

56

5452


143

7.6. Snail graph Sn

Definition 7.6.1

A snail Sn is obtained from path Pn = v1, v2,...,vn by attaching two parallel

edges between vi and vn-i+1 for i = 1, 2, ..., 2

n

.

Theorem 7.6.2

The snail graph Sn (n > 4) is strong edge − graceful for all n is even.

Proof

Let {v1, v2, v3, ..., vn} be the vertices of Sn and {e1, e2, e3, ..., 2 1ne − } be the

edges of Sn which are denoted as in Fig. 7.38.

v1 v2 v3

e1 e2 e3 en-1

vnvn-1

e2n-2

e2n-1

en+1

. . .

. . . . . .

. . .

. . .

. . .

2

nv1

2

nv+

en. . .

Fig. 7.38: Sn with ordinary labeling

144


( )if e = i 1 ≤ i ≤ 2n – 2

( )if e = i + 1 i = 2n – 1


( )1f v+ = n + 1

( )if v+ = n + 2i – 2 2 ≤ i ≤ 2

n

( )if v+ = 2i – n – 2 1 12

ni n+ ≤ ≤ −

( )nf v+ = 2n – 1

Clearly, the vertex labels are all distinct. Hence, the snail graph Sn (n > 4) is

strong edge − graceful for all n is even.

The SEGL of S8 and S12 are illustrated in Fig. 7.39 and Fig. 7.40 respectively.

910 12

1 2 3 6 7154

14

16

9

8

144

11

05

2

13

12

10

Fig. 7.39: S8 with SEGL

145

1314 16 18 20 22 0 2 4 8

231 2 3 4 5 6 7 8 9 10 11

17

16

15

14

13

12

6

18

19

20

21

22

24 Fig. 7.40: S12 with SEGL

Theorem 7.6.3

The snail graph Sn (n ≥ 5) is strong edge − graceful when n is odd.

Proof

Let {v1, v2, v3, ..., vn} be the vertices of Sn and {e1, e2, e3, ..., 2 2ne − } be

the edges of Sn which are denoted as in Fig. 7.41.


( )if e = i 1 ≤ i ≤ 2n – 2


( )if v+ = n – 3 + 2i 1 ≤ i ≤ 1

2

n −

146

1

2

nf v++

= n

( )if v+ = 2i – n – 3 3

12

ni n

+ ≤ ≤ − ; ( )nf v+ = 2n – 3

v1 v2 v3

e1 e2 en-1

e2n-1

e2n-2

en+1

en

. . .

. . .

. . .

1

2

nv + 3

2

nv + 5

2

nv +

. . .

1

2

nv − vn-1 vn

. . .. . .

Fig. 7.41: Sn with ordinary labeling

Clearly, the vertex labels are all distinct. Hence, the snail graph Sn (n ≥ 5) is

strong edge − graceful when n is odd.

The SEGL of S9 and S13 are illustrated in Fig. 7.42 and Fig. 7.43

respectively.

8 10 121 2 3 7 8

154

15

16

10

9

144

12

06

2

14

13

11

59


147

1214 16 18 20 22 0 2 4 8

231 2 3 4 5 6 8 9 10 11 12

18

17

16

15

14

13

6

19

20

21

22

23

24

13

7


CHAPTER – VIII

Strong Edge – Graceful Labeling of Some Disconnected Graphs

148

CHAPTER - VIII


OF SOME DISCONNECTED GRAPHS

8.1. Introduction

In the previous chapter, we have discussed the strong edge – graceful

labeling of several families of graph. In this chapter, we present results on the

strong edge – gracefulness of some disconnected graphs.

� Cm ∪ Pn, m, n ≥ 4, m, n even and m = n.

� Cm ∪ Pn, m, n ≥ 5, m, n odd add m = n.

� Cm ∪ Pn, m ≥ 4, n ≥ 5, m even, n odd and m ≠ n.

� Cm ∪ Pn, m ≥ 3, n ≥ 4, m odd, n even and m ≠ n.

� Cm ∪ Pn, m ≥ 3, n ≥ 5, m, n odd and m ≠ n.

� Cm ∪ Pn, m, n ≥ 4, m, n even and m ≠ n.

� Cm ∪ Cn, m, n ≥ 4, m, n even and m = n.

� Cm ∪ Cn, m, n ≥ 3, m, n odd and m = n.

� Cm ∪ Cn, m ≥ 3, n ≥ 4, m odd, n even and m ≠ n.

� Cm ∪ Cn, m ≥ 4, n ≥ 5, m even, n odd and m ≠ n.

� Cm ∪ Cn, m ≥ 3, n ≥ 5, m, n odd and m ≠ n.

� Cm ∪ Cn, m ≥ 4, n ≥ 6, m, n even and m ≠ n.

� Pm ∪ Pn, m, n ≥ 3, m, n odd and m = n.

� Pm ∪ Pn, m, n ≥ 4, m, n even and m = n.

� Pm ∪ Pn, m, n ≥ 6, m, n even and m ≠ n.

� Pm ∪ Pn, m ≥ 3, n ≥ 5, m, n odd and m ≠ n.

� Pm ∪ Pn, m ≥ 6, n ≥ 5, m even, n odd and m ≠ n.

� Pm ∪ Pn, m ≥ 3, n ≥ 4, m odd, n even and m ≠ n.

149

� Pm ∪ Pn, m = 4, n ≥ 6, m, n even.

� Pm ∪ Pn, m = 4, n ≥ 5, m even, n odd.

� K1,3 ∪ K1,3.

� K1,m ∪ K1,n, m, n ≥ 4, m, n even.

� K1,m ∪ K1,n, m ≥ 3, n ≥ 4, m, n odd.

8.2. Cm ∪∪∪∪ Pn graph

Theorem 8.2.1

The graph Cm ∪ Pn (m, n ≥ 4) is strong edge – graceful for all m, n even

and m = n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of Cm ∪ Pn and

{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '

1ne − } be the edges of Cm ∪ Pn which are denoted

as in the Fig. 8.1.

e3

e2

e1

em

em-1

em-2

vm-1

v1

v2

v3

vm

. . .

. . .

. . .

. . .

. . .

v1' vn'

vn-1'

vn-2'

vn-3'

en-2'

en-1'

en-3'

e2'

e1'

e3'

v2'

v3'

. . .

. . .. . .

. . .

Fig. 8.1: Cm ∪∪∪∪ Pn with ordinary labeling

150

We first label the edges of Cm ∪ Pn as follows:

( )if e = i 1 ≤ i ≤ m – 1

( )mf e = m + 1 ; ( )'1f e = m

( )'1if e+ = m + 1 + i 1 ≤ i ≤ n – 2


( )1f v+ = m + 2

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 2

( )mf v+ = 2m ; ( )'1f v+ = m

( )'2f v+ = 2m + 2

( )'2if v+

+ = 2m + 3+ 2i 1 ≤ i ≤ n – 3

( )'nf v+ = m + n – 1

Clearly, the vertex labels are all distinct. Hence, the graph Cm ∪ Pn

(m, n ≥ 4) is strong edge – graceful for all m, n even and m = n.

The SEGL of C14 ∪ P14 is illustrated in Fig. 8.2.

13 7 15

11

9

7

5

3

16

17

19

21

23

25

28

8

9

10

11

12

13

6

5

4

3

2

1

15

2141 43

39

37

35

33

30

14

45

47

49

51

53

27

14

16

17

18

19

20

27

26

25

24

23

22

Fig. 8.2: C14 ∪∪∪∪ P14 with SEGL

151

Theorem 8.2.2

The graph Cm ∪ Pn (m, n ≥ 5) is strong edge – graceful for all m, n odd

and m = n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '


as in the Fig. 8.1, of Theorem 8.2.1.


( )if e = i 1 ≤ i ≤ m

( )'if e = m + 2 + i 1 ≤ i ≤ n – 2

( )'if e = 2n + 2 i = n – 1


( )1f v+ = m + 1

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 1

( )'1f v+ = n + 3

( )'1if v+

+ = 2m + 5 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 2 ; ( )'nf v+ = 2n + 2


(m, n ≥ 5) is strong edge – graceful for all m, n odd and m = n.

152


13 15 17

11

9

7

5

3

16 15 29

27

25

23

21

19

14

13

12

11

10

9

1

2

3

4

5

6

7 8

18 32

18

37

39

41

43

45

19

20

21

22

23

32

2

30

59

29

57

28

55

27

53

26

4724

4925

51


Theorem 8.2.3

The graph Cm ∪ Pn (m ≥ 4, n ≥ 5) is strong edge – graceful for all

m even, n odd and m ≠ n, except when n – m = 1, 3.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 1

( )mf e = m + 1 ; ( )'1f e = m

( )'1if e+ = m + 1 + i 1 ≤ i ≤ n - 2

153


( )1f v+ = m + 2

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 2

( )mf v+ = 2m ; ( )'1f v+ = m

( )'2f v+ = 2m + 2

( )'2if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 3

( )'nf v+ = m + n – 1


(m ≥ 4, n ≥ 5) is strong edge – graceful for all m even, n odd and m ≠ n, except

when n – m = 1, 3.


11 6 13

9

7

5

3

14 2413

11

21

19

9

8

17

15

10

7

1

2

3

4

5

53

26

51

25

49

24

47

23

45

22

43

21

4120

39

19

37

18

35

33

17

31

29

26

12

16

15

14

1227

55

28

28


154

Theorem 8.2.4

The graph Cm ∪ Pn (m ≥ 3, n ≥ 4) is strong edge – graceful for all m odd,

n even and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m

( )'if e = m + 2 + i 1 ≤ i ≤ n – 1


( )1f v+ = m + 1

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 1

( )'1f v+ = m + 3

( )'1if v+

+ = 2m + 5 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 1 ; ( )'nf v+ = m + n + 1


(m ≥ 3, n ≥ 4) is strong edge – graceful for all m odd, n even and m ≠ n.

The SEGL of C13 ∪ P10 and C11 ∪ P14 are illustrated in Fig. 8.5 and


155

14 13 25

23

21

19

17

151311

9

7

5

3

1

2

3

4

56 7

8

9

10

11

12

2039 41

43

45

1

37

35

33

16 24

16

17

18

19 21

22

23

24


12 21

1

3

5

7

9 13115 6

4

3

2

15

17

19

7

8

9

10

11

39 20 41

37

35

33

31

29

14 26

1

49

47

45

43

19

18

17

16

15

14

21

22

23

24

25

26


156

Theorem 8.2.5

The graph Cm ∪ Pn (m ≥ 3, n ≥ 5) is strong edge – graceful for all m, n

odd and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m

( )'if e = m + 2 + i 1 ≤ i ≤ n – 2

( )'1nf e − = m + n + 2


( )1f v+ = m + 1

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 1

( )'1f v+ = m + 3

( )'1if v+

+ = 2m + 5 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 2 ; ( )'nf v+ = m + n + 2


(m ≥ 3, n ≥ 5) is strong edge – graceful for all m, n odd and m ≠ n.



157

12 11 21

1

3

5

7 15

17

19

2

3

4

9 11 13

5 67

8

9

10

14 28

29

31

33

35

37

39 41 43

45

47

49

51

2

19

18

17

16

15

14

22

23

24

25

26

28

20 21


18 33

31

29

27

25

23

21

191715

13

11

9

7

5

3

8 97

6

5

4

3

2

1 16

15

14

13

12

11

10

17

20 32

41

43

45

47

49 51 53

55

57

59

2

25 26

24

23

22

21

20

27

28

29

30

32


158

Theorem 8.2.6

The graph Cm ∪ Pn (m, n ≥ 4) is strong edge – graceful for all m, n even

and m ≠ n, except when n – m = 2.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 1

( )mf e = m + 1 ; ( )'1f e = m

( )'1if e+ = m + 1 + i 1 ≤ i ≤ n – 3

( )'1nf e − = m + n


( )1f v+ = m + 2

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 2

( )mf v+ = 2m ; ( )'1f v+ = m

( )'2f v+ = 2m + 2

( )'2if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 4

( )'1nf v+

− = 2m + 2n – 2 ; ( )'nf v+ = m + n

159

Clearly, the vertex labels are all distinct. Hence, the graph Cm ∪ Pn (m, n ≥ 4)

is strong edge – graceful for all m, n even and m ≠ n, except when n – m = 2.



11 6 13

9

7

5

3

15

17

19

21

14 2413

5

4

3

2

1

7

8

9

10

11

12 30

26

29

31

33

35

37

39

41 4321

45

47

49

51

53

55

58

20

19

18

17

16

15

14

12 30

28

27

26

25

24

23

22


160

57

66

22 4021

3

5

7

9

11

13

15

17

19 2110

23

25

27

29

31

33

35

37

1

2

3

4

6

7

8

5

9

19

18

17

16

15

14

13

12

11

20 34

42

45

47

49

51

53 5527

59

61

63

34

32

31

30

29

2826

25

24

23

22

20

Fig. 8.10: C20 ∪∪∪∪ P14 with SEGL

8.3. Cm ∪∪∪∪ Cn graph

Theorem 8.3.1

The graph Cm ∪ Cn (m, n ≥ 4) is strong edge – graceful for all m, n even

and m = n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of Cm ∪ Cn and

{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '

ne } be the edges of Cm ∪ Cn which are denoted

as in the Fig. 8.11.

161

v4

v3

v2

v1 em vm

e3

e2

e1 em-1

em-2

em-3

vm-1

vm-2

vm-3v4'

v3'

v2'

v1' en' vn'

e3'

e2'

e1' en-1'

en-2'

en-3'

vn-1'

vn-2'

vn-3'

.

.

.

...

.

.

.

.

.

.

.

.

.

...

Fig. 8.11: Cm ∪∪∪∪ Cn with ordinary labeling

We first label the edges of Cm ∪ Cn as follows:

( )if e = i 1 ≤ i ≤ m – 1

( )mf e = m + 1 ; ( )'1f e = m

( )'1if e+ = m + 1 + i 1 ≤ i ≤ n – 1


( )1f v+ = m +2

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 2

( )mf v+ = 2m ; ( )'1f v+ = 3m

( )'2f v+ = 2m + 2

( )'2if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 2

162

Clearly, the vertex labels are all distinct. Hence, the graph Cm ∪ Cn


The SEGL of C16 ∪ C16 is illustrated in Fig. 8.12.

15 8 17

13

11

9

7

5

3

18 17 32

29

27

25

23

21

19

7

6

5

4

3

2

1 15

14

13

12

11

10

9

48 32 63

34

37

39

41

43

45

61

59

57

55

53

51

47 4924

16

18

19

20

21

22

31

30

29

28

27

26

23 25

Fig. 8.12: C16 ∪∪∪∪ C16 with SEGL

Theorem 8.3.2

The graph Cm ∪ Cn (m, n ≥ 3) is strong edge – graceful for all m, n odd

and m = n.

Proof

Let{v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m

163

( )'if e = m + i 1 ≤ i ≤ n


( )1f v+ = m + 1

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 1

( )'1f v+ = 3n + 1

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 1



The SEGL of C21 ∪ C21 is illustrated in Fig. 8.13.

22 4121

3

5

7

9

11

13

15

17

19 21 23

10 11

25

27

29

31

33

35

37

39

20

19

18

17

16

15

14

13

129

8

7

6

5

4

3

2

1

64 42 83

45

47

49

51

53

55

81

79

77

75

73

71

57

59

61 63 65

67

69

31 32

30 33

29 34

28 35

27 36

26 37

25 38

24 39

23 40

22 41

Fig. 8.13: C21 ∪∪∪∪ C21 with SEGL

Theorem 8.3.3

The graph Cm ∪ Cn (m ≥ 3, n ≥ 4) is strong edge – graceful for all m odd,

n even and m ≠ n.

164

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m

( )'if e = m + 2 + i 1 ≤ i ≤ n – 1

( )'nf e = m + n + 3


( )1f v+ = m + 1

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 1

( )'1f v+ =

2 if 4

0 if 6

2 6 if 8

n

n

m n n

= = + + ≥

( )'1if v+

+ = 2m + 5 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 1

( )'nf v+ = 4



The SEGL of C15 ∪ C10 and C13 ∪ C16 are illustrated in Fig. 8.14 and


165

13 15 17

11

9

7

5

3

16 15 29

27

25

23

21

19

6

7 8

9

5 10

4

3

2

1

11

12

13

14

43 22 45

21

41

20

39

19

37

18

46 28 4

1

49

47

23

24

25

26

Fig. 8.14: C15 ∪∪∪∪ C10 with SEGL

11

6

137

15

5

9

7

5

3

17

19

21

23

14 2513

1

2

3

4

8

9

10

11

12

48 32 4

33

35

37

39

41

43

45 23 47

49

51

53

55

57

1

16

17

18

19

20

21

22 24

25

26

27

28

29

30

Fig. 8.15: C13 ∪∪∪∪ C16 with SEGL

166

Theorem 8.3.4

The graph Cm ∪ Cn (m ≥ 4, n ≥ 5) is strong edge – graceful for all m

even, n odd and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 1

( )mf e = i + 1 ; ( )'1f e = m

( )'1if e+ = m + 1 + i 1 ≤ i ≤ n – 2

( )'nf e = m + n + 1


( )1f v+ = m + 2

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 2

( )mf v+ = 2m ; ( )'1f v+ = 2m + n + 1

( )'2f v+ = 2m + 2

( )2if v++ = 2m + 3 + 2i 1 ≤ i ≤ n – 3

( )'nf v+ = 0


(m ≥ 4, n ≥ 5) is strong edge – graceful for all m even, n odd and m ≠ n.

167



16 2815

3

5

7

9

11

13 7 15

17

19

21

23

25

1

2

3

4

5

6 8

9

10

11

12

13

37

19

39

20

41

35

33

30

43

45

47

040 26

18

17

16

14

21

22

23

24

Fig. 8.16: C14 ∪∪∪∪ C11 with SEGL

11 6 13

9

7

5

3

14 13 24

21

19

17

15

5

4

3

2

1

7

8

9

10

11

41

21

43

22

45

47

49

51

53

55

57

59

03244

26

29

31

33

35

37

39

20

19

18

17

16

15

14

12 30

29

28

27

26

25

24

23

Fig. 8.17: C12 ∪∪∪∪ C19 with SEGL

168

Theorem 8.3.5

The graph Cm ∪ Cn (m ≥ 3, n ≥ 5) is strong edge – graceful for all m, n

odd and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m

( )'if e = m + i 1 ≤ i ≤ n


( )1f v+ = m + 1

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 1

( )'1f v+ = 2m + n + 1

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 1





169

13 15 17

19

21

23

25

27

29

11

9

7

5

3

16 15

6

7 8

5

4

3

2

1

9

10

11

12

13

14

2642 51

49

47

45

434139

20 21

37

35

33

19

18

17

16

22

23

24

25

Fig. 8.18: C15 ∪∪∪∪ C11 with SEGL

11 13 15

17

19

21

23

25

9

7

5

3

14

6 7

8

9

10

11

12

5

4

3

2

1

13

45 47 49

43

41

39

37

35

33

31

29

48 67

65

63

61

59

57

55

53

51

23 24

22 25

21 26

20 27

19 28

18 29

17 30

16 31

15 32

14 33

34 Fig. 8.19: C13 ∪∪∪∪ C21 with SEGL

170

Theorem 8.3.6

The graph Cm ∪ Cn (m ≥ 4, n ≥ 6) is strong edge – graceful for all m, n

even and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 1

( )mf e = m + 1 ; ( )'1f e = m

( )'1if e+ = m + 1 + i 1 ≤ i ≤ n – 1


( )1f v+ = m + 2

( )1if v++ = 2i + 1 1 ≤ i ≤ m – 2

( )mf v+ = 2m ; ( )'1f v+ = 2m + n

( )'2f v+ = 2m + 2

( )'2if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 2


(m ≥ 4, n ≥ 6) is strong edge – graceful for all m, n even and m ≠ n.



171

11 6 13

15

17

19

21

2414 13

3

5

7

9

5

4

3

2

1

7

8

9

10

11

39 20 41

37

35

33

31

29

26

40 5528

53

51

49

47

45

43

12

14

15

16

17

18

19 21

22

23

24

25

26

27

Fig. 8.20: C12 ∪∪∪∪ C16 with SEGL

20 19 36

33

31

29

27

25

23

21

1917

15

13

11

9

7

5

3

9

8

7

6

5

4

3

2

1 17

16

15

14

13

12

11

10

47 24 49

45

43

41

38

48

51

53

55

57

5930

18

20

21

22

23 25

26

27

28

29

Fig. 8.21: C18 ∪∪∪∪ C12 with SEGL

172

8.4. Pm ∪∪∪∪ Pn graph

Theorem 8.4.1

The graph Pm ∪ Pn (m, n ≥ 3) is strong edge – graceful for all m, n odd

and m = n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '

nv } be the vertices of Pm ∪ Pn and

{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '

1ne − } be the edges of Pm ∪ Pn which are

denoted as in the Fig. 8.22.

v1 v2 v3

v4

vm-2 vm-1 vm

vm-3

em-2 em-1

em-3

e2e1

e3

v1' v2' v3'

v4'

vn-2' vn-1' vn'

vn-3'

en-2' en-1'en-3'

e2'e1'e3'

.

.

.

.

.

.

...

...

.

.

.

.

.

.

...

...

.

.

.

.

.

.

.

.

.

.

.

.

Fig. 8.22: Pm ∪∪∪∪ Pn with ordinary labeling

We first label the edges of Pm ∪ Pn as follows:

( )if e = i 1 ≤ i ≤ m – 1

( )'if e = m + i 1 ≤ i ≤ n – 2

( )'1nf e − = 2n


( )if v+ = 2i – 1 1 ≤ i ≤ m – 1

( )mf v+ = m – 1 ; ( )'1f v+ = m + 1

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 4n – 2 ; ( )'nf v+ = 2n

173

Clearly, the vertex labels are all distinct. Hence, the graph Pm ∪ Pn


The SEGL of P15 ∪ P15 is illustrated in Fig. 8.23.

1 3

7

9

5

11

13 15 17

19

21

23

25 27 14

7 86 9

5 10

4 11

3 12

1 2 13 14

16 33 35

37

39

41

43 45 47

49

51

53

55 58 30

22 2321 24

20 25

19 26

18 27

16 17 28 30

Fig. 8.23: P15 ∪∪∪∪ P15 with SEGL

Theorem 8.4.2

The graph Pm ∪ Pn (m, n ≥ 4) is strong edge – graceful for all m, n even

and m = n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 2

( )if e = i + 1 i = m – 1

( )'if e = m + 1 + i 1 ≤ i ≤ n – 1


( )if v+ = 2i – 1 1 ≤ i ≤ m – 2

174

( )1mf v+− = 2m – 2 ; ( )mf v+ = m

( )'1f v+ = m + 2

( )'1if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 2

( )'nf v+ = m + n




158

17

13

11

9

7

5

3

1

19

21

23

25

27

30

16

7

6

5

4

3

2

1

9

10

11

12

13

14

16

49 5125

47

45

43

41

39

37

18

53

55

57

59

61

63

32

26

27

28

29

30

31

32

24

23

22

21

20

19

18

Fig. 8.24: P16 ∪∪∪∪ P16 with SEGL

175

Theorem 8.4.3

The graph Pm ∪ Pn (m, n ≥ 6) is strong edge – graceful for all m, n even

and m ≠ n, except when m – n = 2.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 2

( )if e = i + 1 i = m – 1

( )'if e = m + 1 + i 1 ≤ i ≤ n – 1


( )if v+ = 2i – 1 1 ≤ i ≤ m – 2

( )1mf v+− = 2m – 2 ; ( )mf v+ = m

( )'1f v+ = m + 2

( )'1if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 2

( )'nf v+ = m + n


(m, n ≥ 6) is strong edge – graceful for all m, n even and m ≠ n, except when

m – n = 2.

The SEGL of P14 ∪ P20 and P22 ∪ P12 are illustrated in Fig. 8.25 and


176

13 7 15

11

9

7

5

3

1

17

19

21

23

26

14

6

5

4

3

2

1

8

9

10

11

12

14

67

65

63

61

59

57

55

53

16 34

512549

47

45

43

41

39

37

35

33

24

23

22

21

20

19

18

26

27

28

29

30

31

32

17

16

33

34

Fig. 8.25: P14 ∪∪∪∪ P20 with SEGL

21 11 23

19

17

15

13

11

9

7

5

3 42

39

37

35

33

31

29

27

25

10

9

8

7

6

5

4

3

2

1 22

20

1 22

19

18

17

16

15

14

13

12

57 29 59

55

53

51

49

24 34

67

65

63

61

24

25

26

27

28 30

31

32

33

34

Fig. 8.26: P22 ∪∪∪∪ P12 with SEGL

177

Theorem 8.4.4

The graph Pm ∪ Pn (m ≥ 3, n ≥ 5) is strong edge – graceful for all m, n

odd and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 1

( )'if e = m + i 1 ≤ i ≤ n – 2

( )'1nf e − = m + n


( )if v+ = 2i – 1 1 ≤ i ≤ m – 1

( )mf v+ = m – 1 ; ( )'1f v+ = m + 1

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 2m + 2n – 2 ; ( )'nf v+ = m + n





178

13 15 17

11

9

7

5

3

1

19

21

23

25

27

14

7 8

6 9

5 10

4 11

12

13

14

3

2

1

47 49 51

45

43

41

39

37

35

33

16 34

66

63

61

59

57

55

53

24 2523 26

22 27

21 28

20 29

19 30

18 31

17 32

16 34

Fig. 8.27: P15 ∪∪∪∪ P19 with SEGL

19 21 23

17

15

13

11

9

7

5

3

1

25

27

29

31

33

35

37

39

20

20

19

18

17

16

15

14

13

129

8

7

6

5

4

3

2

1

10 1153 55 57

51

49

47

45

22

59

61

63

66

34

22

23

24

25

2627 28

29

30

31

32

34

Fig. 8.28: P21 ∪∪∪∪ P13 with SEGL

179

Theorem 8.4.5

The graph Pm ∪ Pn (m ≥ 6, n ≥ 5) is strong edge – graceful for all m

even, n odd and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 2

( )if e = i + 1 i = m – 1

( )'if e = m + 1 + i 1 ≤ i ≤ n – 2

( )'1nf e − = m + n + 1


( )if v+ = 2i – 1 1 ≤ i ≤ m – 2

( )1mf v+− = 2m – 2 ; ( )mf v+ = m

( )'1f v+ = m + 2

( )'1if v+

+ = 2m + 3 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 0 ; ( )'nf v+ = m + n + 1


(m ≥ 6, n ≥ 5) is strong edge – graceful for all m even, n odd and m ≠ n.



180

1

3

5

77

9

11

13

7

15

17

19

21

23

26

14

14

12

11

10

9

86

5

4

3

2

1

16 36

33

35

37

39

41

43

45

47

49 51 53

55

57

59

61

63

65

67

0

36

34

33

32

31

30

29

28

27

2625

24

23

22

21

20

19

18

17

16

Fig. 8.29: P14 ∪∪∪∪ P21 with SEGL

19

10

21

17

15

13

11

9

7

5

3

1 20

38

35

33

31

29

27

25

23

9

8

7

6

5

4

3

2

1 20

18

17

16

15

14

13

12

11

51 53 55

49

47

45

22

57

59

0

32

26 27

25

24

23

22

28

29

30

32

Fig. 8.30: P20 ∪∪∪∪ P11 with SEGL

181

Theorem 8.4.6

The graph Pm ∪ Pn (m ≥ 3, n ≥ 4) is strong edge – graceful for all m odd,

n even and m ≠ n.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ m – 1

( )'if e = m + i 1 ≤ i ≤ n – 1


( )if v+ = 2i – 1 1 ≤ i ≤ m – 1

( )mf v+ = m – 1 ; ( )'1f v+ = m + 1

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 2

( )'nf v+ = m + n – 1





182

13 15 17

11

9

7

5

3

1

19

21

23

25

27

14

14

13

12

11

10

96

5

4

3

2

1

7 8

16

33

35

37

39

41

43

45

47

49

51

53

27

55

57

59

61

63

65

67

69

71

73

75

38

16

17

18

19

20

21

22

23

24

25

26 28

29

30

31

32

33

34

35

36

37

38

Fig. 8.31: P15 ∪∪∪∪ P24 with SEGL

15 17 19

13

11

9

7

5

3

1

21

23

25

27

29

31

16

16

15

14

13

12

11

107

6

5

4

3

2

1

8 9 43

2245

41

39

37

18

47

49

51

26

21

20

19

18

23

24

25

26

Fig. 8.32: P17 ∪∪∪∪ P10 with SEGL

183

Theorem 8.4.7

The graph Pm ∪ Pn (m = 4, n ≥ 6) is strong edge – graceful for all n

even.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ 2

( )if e = i + 1 i = 3

( )'if e = m + i 1 ≤ i ≤ n – 2

( )'1nf e − = m + n


( )if v+ = 2i – 1 1 ≤ i ≤ 2

( )3f v+ = m + 2 ; ( )4f v+ = m

( )'1f v+ = 5

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 3

( )'1nf v+

− = 2m + 2n – 2 ; ( )'nf v+ = m + n


(m = 4, n ≥ 6) is strong edge – graceful for all n even.


184

1

1 3

2

6

4

4

12

23 25

27

29

31

33

35 38 20

21

19

17

15

13115

5 67

8

9

10

11 13

14

15

16

1718 20

Fig. 8.33: P4 ∪∪∪∪ P16 with SEGL

Theorem 8.4.8

The graph Pm ∪ Pn (m = 4, n ≥ 5) is strong edge – graceful for all

n odd.

Proof

Let {v1, v2, v3, ..., vm, ' ' '1 2 3, , v v v , ..., '


{ e1, e2, e3, ..., em-1, ' ' '1 2 3, , e e e, ..., '




( )if e = i 1 ≤ i ≤ 2

( )if e = i + 1 i = 3

( )'if e = m + i 1 ≤ i ≤ n – 1


( )if v+ = 2i – 1 1 ≤ i ≤ 2

( )3f v+ = m + 2 ; ( )4f v+ = m

185

( )'1f v+ = 5

( )'1if v+

+ = 2m + 1 + 2i 1 ≤ i ≤ n – 2

( )'nf v+ = m + n – 1


(m = 4, n ≥ 5) is strong edge – graceful for all n odd.


1

1 3

2

6

4

4

19 21 23

17

15

13115

25

27

29 31 16

121110

9

8

765

13

1415 16

Fig. 8.34: P4 ∪∪∪∪ P13 with SEGL

8.5. K1,m ∪∪∪∪ K1,n graph

Theorem 8.5.1

The graph K1,3 ∪ K1,3 is strong edge – graceful.

Proof

The strong edge – graceful labeling of K1,3 ∪ K1,3 is shown in Fig. 8.35.

7

1 2 4

1 2 4

3

5 6 8

5 6 8

Fig. 8.35: K1,3 ∪∪∪∪ K1,3 with SEGL

Here p = m + n + 2 = 8 and 2p = 16 ; 3

= 92

q.

186

Theorem 8.5.2

The graph K1,m ∪ K1,n (m, n ≥ 4) is strong edge – graceful for all m, n

even.

Proof

Let { }' ' '0 1 0 1, , ..., , , , ..., m nv v v v v v be the vertices of K1,m ∪ K1,n and

{ }' ' '1 2 1 2, , ..., , , , ..., m ne e e e e e be the edges of K1, m ∪ K1, n which are denoted in

the Fig. 8.36.

v0

v1 v2 vm

...e1 e2 em

v0'

v1' v2

' vn'

e1'

e2' en

'

Fig. 8.36: K1,m ∪∪∪∪ K1,n with ordinary labeling

Consider the Diophantine equation x1 + x2 = 2p and its solutions of the

form (t, 2p – t) where 2p − 3

2

q

≤ t ≤ 2p − 3

2

q

+ 6

2

m n+ −. There will be

4

2

m n+ − pair of solutions. (i.e)

2

2

m− +

2

2

n − pair of solutions.

With there pairs, label the edges {ei : 3 ≤ i ≤ m} of K1,m and the edges

{ 'ie : 3 ≤ i ≤ n} of K1,n by the coordinates of the pair in any order so that

adjacent edges receive the coordinates of the pairs.

Now, we label the remaining edges as follows:

( )1f e = 1 ; ( )2f e = 6

( )'1f e = 2 ; ( )'

2f e = 3

187


( )0f v+ = 7 ; ( )'0f v+ = 5.

And all the pendant vertices will receive labels of the edges with which

they are incident and they are distinct.

Hence the graph K1,m ∪ K1,n is strong edge – graceful for all m, n even.

The SEGL of K1,8 ∪ K1,8 is given in Fig. 8.37.

7

1 6 24 12 23 13 22 14

16 24 12 23 13 22 14

5

2 3 21 15 20 16 19 17

23 21 15 20 16 19 17

Fig. 8.37: K1,8 ∪∪∪∪ K1,8 with SEGL

The SEGL of K1,10 ∪ K1,14 is given in Fig. 8.38.

7

1 6 36 16 35 17 34 18

6 36 16 35 17 34 18 33

33 19

119

2 3 32 20 31 21 30 22 29 23 28 24 27 25

2 3 32 20 31 21 30 22 29 23 28 24 27 25

5

Fig. 8.38: K1,10 ∪∪∪∪ K1,14 with SEGL

Theorem 8.5.3

The graph K1,m ∪ K1,n (m ≥ 3, n ≥ 4) is strong edge − graceful for all m,

n odd.

188

Proof

Let { }' ' '0 1 0 1, , ..., , , , ..., m nv v v v v v be the vertices of K1,m ∪ K1,n and

{ }' ' '1 2 1 2, , ..., , , , ..., m ne e e e e e be the edges of K1,m ∪ K1,n which are denoted in

the fig. 8.36, of theorem 8.5.2.

Consider the Diophantine equation x1 + x2 = 2p and its solutions of the

form (t, 2p – t) where 2p − 3

2

q

≤ t ≤ 2p − 3

2

q

+ 8

2

m n+ −. There will be

6

2

m n+ − =

3

2

m− +

3

2

n − pair of solutions.

With the above pairs, (except p – 2 = 14 or p + 2 = 14 as one of the

coordinates) label the edges {ei : 4 ≤ i ≤ m} of K1,m and the edges {'ie : 4 ≤ i ≤ n}

of K1,n by the coordinates of the pair in any order so that adjacent edges receive

the coordinates of the pairs.

Now, we label the remaining edges as follows:

( )1f e = 1 ; ( )2f e = 2

( )3f e = 4 ; ( )'1f e = 3

( )'2f e = 5 ; ( )'

3f e = 6


( )0f v+ = 7 ; ( )'0f v+ = 14.

And all the pendant vertices will receive labels of the edges with which

they are incident and they are distinct.

Hence the graph K1,m ∪ K1,n are strong edge – graceful for all m, n odd.

189

The SEGL of K1,7 ∪ K1,7 and K1,5 ∪ K1,9 are given in Fig. 8.39 and


7

1 2 4 11 21 12 20

12 4 11 21 12 20

14

3 5 6 13 19 17 15

35 6 13 19 17 15

Fig. 8.39: K1,7 ∪∪∪∪ K1,7 with SEGL

1 2 4 21 11

1 2 4 21 11

7

3 5 6 20 12 19 13 17 15

35 6 20 12 19 13 17 15

14

Fig. 8.40: K1,5 ∪∪∪∪ K1,9 with SEGL

Bibliography

190

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Documents

A STUDY ON SOME VARIATIONS OF GRAPH LABELING AND ITS