ODD SUM LABELING OF SOME SUBDIVISION GRAPHS 1

Kragujevac Journal of Mathematics

Volume 38(1) (2014), Pages 203–222.

ODD SUM LABELING OF SOME SUBDIVISION GRAPHS

S. AROCKIARAJ1, P. MAHALAKSHMI2, AND P. NAMASIVAYAM3

Abstract. An injective function f : V (G)→ {0, 1, 2, . . . , q} is an odd sum labelingif the induced edge labeling f∗ defined by f∗(uv) = f(u) + f(v), for all uv ∈ E(G),is bijective and f∗(E(G)) = {1, 3, 5, . . . , 2q − 1}. A graph is said to be an odd sumgraph if it admits an odd sum labeling. In this paper, we have studied the odd sumproperty of the subdivision of the triangular snake, quadrilateral snake, slantingladder, Cp�K1, H �K1, Cm@Cn, the grid graph Pm×Pn, duplication of a vertexof a path and duplication of a vertex of a cycle.

1. Introduction

Throughout this paper, by a graph, we mean a finite, undirected simple graph. LetG(V,E) be a graph with p vertices and q edges. For notations and terminology, wefollow [1].

A path on p vertices is denoted by Pp and a cycle on p vertices is denoted by Cp.If m number of pendant vertices are attached at each vertex of G, then the resultantgraph obtained from G is the graph G �mK1. When m = 1, G � K1 is the coronaof G. A triangular (quadrilateral) snake is obtained from a path by identifying eachedge of the path with an edge of the cycle C3(C4). The graph Cm@Cn is obtainedby identifying an edge of Cm with an edge of Cn. A graph which can be obtainedfrom a given graph by breaking up each edge into one or more segments by insertingintermediate vertices between its two ends. If each edge of a graph G is broken intotwo by exactly one vertex, then the resultant graph is taken as S(G). The slantingladder SLn is a graph obtained from two paths u1u2 . . . un and v1v2 . . . vn by joiningeach ui with vi+1, 1 ≤ i ≤ n− 1.

Key words and phrases. Odd sum labeling, Odd sum graphs.2010 Mathematics Subject Classification. 05C78.Received: August 3, 2013Revised: April 15, 2014.

203

204 S. AROCKIARAJ, P. MAHALAKSHMI, AND P. NAMASIVAYAM

Duplication of a vertex v of graph G produces a new graph G′ by adding a newvertex v′ such that N(v′) = N(v). In other words, a vertex v′ is said to be duplicationof v if all the vertices which are adjacent to v in G are also adjacent to v′ in G′.

In [2], an odd edge labeling of a graph is defined as follows: A labeling f : V (G)→{0, 1, 2, . . . , p − 1} is called an odd edge labeling of G if for the edge labeling f+ onE(G) defined by f+(uv) = f(u) + f(v) for any edge uv ∈ E(G), for a connectedgraph G, the edge labeling is not necessarily injective. In [5], the concept of pair sumlabeling was introduced. An injective function f : V (G) → {±1,±2, . . . ,±p} is saidto be a pair sum labeling if the induced edge function fe : E(G)→ Z−{0} defined by

fe(uv) = f(u)+f(v) is one-one and fe(E(G)) is either of the form {±k1,±k2, . . . ,±kq2}

or {±k1,±k2, . . . ,±kq−1

2} ∪ {kq+1

2} according as q is even or odd. A graph with a pair

sum labeling defined on it is called a pair sum graph. In [6], the concept of meanlabeling was introduced. An injective function f : V (G) → {0, 1, 2, . . . , q} is said tobe a mean labeling if the induced edge labeling f ∗ defined by

f ∗(uv) =

{ f(u)+f(v)2

, if f(u) + f(v) is evenf(u)+f(v)+1

2, if f(u) + f(v) is odd

is injective and f ∗(E(G)) = {1, 2, . . . , q}. A graph G is said to be odd mean graph ifthere exists an injective function f from V (G) to {0, 1, 2, 3, . . . , 2q− 1} such that theinduced map f ∗ from E(G)→ {1, 3, 5, . . . , 2q − 1} defined by

f ∗(uv) =

{ f(u)+f(v)2

, if f(u) + f(v) is evenf(u)+f(v)+1

2, if f(u) + f(v) is odd

is a bijection [4].Motivated by these, we introduce a new concept called odd sum labeling. An

injective function f : V (G) → {0, 1, 2, . . . , q} is an odd sum labeling if the inducededge labeling f ∗ defined by f ∗(uv) = f(u) + f(v), for all uv ∈ E(G) is bijective andf ∗(E(G)) = {1, 3, 5, . . . , 2q− 1}. A graph is said to be an odd sum graph if it admitsan odd sum labeling.

In this paper, we have studied the odd sum property of the subdivision of thetriangular snake, quadrilateral snake, slanting ladder, Cp �K1, H �K1, Cm@Cn, thegrid graph Pm × Pn, duplication of a vertex of path and cycle by a vertex.

2. Main Results

Proposition 2.1. S(Tn) is an odd sum graph, when n is even.

Proof. Let u1, u2, . . . , un, un+1 the vertices on the path of length n in Tn and letvi, 1 ≤ i ≤ n be the vertices of Tn in which vi is adjacent to ui and ui+1. Let xi, yiand zi be the vertices which subdivide the edges uiui+1, uivi and viui+1 respectivelyfor each i, 1 ≤ i ≤ n.

ODD SUM LABELING OF SOME SUBDIVISION GRAPHS 205

Define f : V (S(Tn))→ {0, 1, 2, . . . , 6n} as follows:

f(ui) = 6i− 6, 1 ≤ i ≤ n + 1

f(vi) =

{6i− 4, 1 ≤ i ≤ n and i is odd6i− 2, 1 ≤ i ≤ n and i is even

f(xi) =

{6i + 1, 1 ≤ i ≤ n and i is odd6i− 7, 1 ≤ i ≤ n and i is even

f(yi) =


and f(zi) =

{6i− 3, 1 ≤ i ≤ n and i is odd6i− 1, 1 ≤ i ≤ n and i is even.

Then the induced edge labeling is obtained as follows:

f ∗(uixi) =


f ∗(xiui+1) =


f ∗(uiyi) =


f ∗(yivi) =


f ∗(vizi) =


and f ∗(ziui+1) =

{12i− 3, 1 ≤ i ≤ n and i is odd12i− 1, 1 ≤ i ≤ n and i is even.

Thus, f is an odd sum labeling of S(Tn) where n is even. �

Proposition 2.2. The graph S(Qn) is an odd sum graph for any n.

Proof. Let u1, u2, . . . , un, un+1 be the vertices on the path of length n in Qn and letvi and wi be the vertices of Qn in which vi is adjacent to ui and wi is adjacent toui+1, for each i, 1 ≤ i ≤ n. Let ti, xi, yi, zi be the vertices which subdivide the edgesuiui+1, uivi, viwi and wiui+1 respectively for each i, 1 ≤ i ≤ n.

We define f : V (S(Qn))→ {0, 1, 2, . . . , 8n} as follows: f(ui) = 8i−8, 1 ≤ i ≤ n+1,f(vi) = 8i − 6, 1 ≤ i ≤ n f(wi) = 8i − 2, 1 ≤ i ≤ n, f(ti) = 8i − 1, 1 ≤ i ≤ n,f(xi) = 8i − 7, 1 ≤ i ≤ n f(yi) = 8i − 5, 1 ≤ i ≤ n and f(zi) = 8i − 3, 1 ≤ i ≤ n.Then the induced edge labeling is obtained as follows: f ∗(uixi) = 16i−15, 1 ≤ i ≤ n,f ∗(xivi) = 16i − 13, 1 ≤ i ≤ n, f ∗(viyi) = 16i − 11, 1 ≤ i ≤ n, f ∗(yiwi) = 16i − 7,1 ≤ i ≤ n, f ∗(wizi) = 16i − 5, 1 ≤ i ≤ n, f ∗(ziui+1) = 16i − 3, 1 ≤ i ≤ n,f ∗(uiti) = 16i− 9, 1 ≤ i ≤ n and f ∗(tiui+1) = 16i− 1, 1 ≤ i ≤ n.

Thus, f is an odd sum labeling of S(Qn). �


Proposition 2.3. The subdivision graph of slanting ladder S(SLn) is an odd sumgraph.

Proof. Let u1, u2, . . . , un and v1, v2, . . . , vn be the vertices on the paths of length n−1. Let xi, yi and zi be the vertices subdivided the edges uiui+1, vivi+1 and uivi+1

respectively for each i, 1 ≤ i ≤ n− 1.Case (i). n is even.

We define f : V (S(SLn))→ {1, 2, . . . , 6(n− 1)} as follows:

f(ui) =

6i− 2, 1 ≤ i ≤ n and i is odd6i− 4, 1 ≤ i ≤ n− 2 and i is even6(n− 1), i = n

f(vi) =

0, i = 16i− 10, 2 ≤ i ≤ n and i is even6i− 8, 2 ≤ i ≤ n and i is odd

f(xi) =

{6i− 1, 1 ≤ i ≤ n− 1 and i is odd6i + 3, 1 ≤ i ≤ n− 1 and i is even

f(yi) =

1, i = 16i− 3, 2 ≤ i ≤ n− 1 and i is even6i− 7, 2 ≤ i ≤ n− 1 and i is odd

and

f(zi) =

{3, i = 16i− 5, 2 ≤ i ≤ n− 1.


f ∗(uixi) =

{12i− 13, 1 ≤ i ≤ n− 1 and i is odd12i− 1, 1 ≤ i ≤ n− 1 and i is even

f ∗(xiui+1) =

12i + 1, 1 ≤ i ≤ n− 2 and i is odd12i + 7, 1 ≤ i ≤ n− 2 and i is even12n− 13, i = n− 1

f ∗(viyi) =


f ∗(yivi+1) =


f ∗(uizi) =



and f ∗(zivi+1) =

5, i = 112i− 9, 2 ≤ i ≤ n− 1 and i is odd12i− 7, 2 ≤ i ≤ n− 1 and i is even.

Thus, f is an odd sum labeling of S(SLn), for n ≥ 4.When n = 2, an odd sum labeling of the graph is given below.

Figure 1. S(SL2)

Case (ii). n is odd.We define f : V (SLn)→ {1, 2, . . . , 6(n− 1)} as follows:

f(ui) =

4, i = 112, i = 26i− 4, 3 ≤ i ≤ n− 1 and i is odd6i− 2, 3 ≤ i ≤ n− 1 and i is even6(n− 1), i = n

f(vi) =

0, i = 12, i = 26, i = 36i− 8, 4 ≤ i ≤ n and i is even6i− 10, 4 ≤ i ≤ n and i is odd

f(xi) =

5, i = 16i− 1, 2 ≤ i ≤ n− 1 and i is even6i + 3, 2 ≤ i ≤ n− 1 and i is odd

f(yi) =

1, i = 19, i = 26i− 3, 3 ≤ i ≤ n− 1 and i is odd6i− 7, 3 ≤ i ≤ n− 1 and i is even

and f(zi) =

{3, i = 16i− 5, 2 ≤ i ≤ n− 1.


The induced edge labeling is obtained as follows:

f ∗(uixi) =


f ∗(xiui+1) =

17, i = 112i + 1, 2 ≤ i ≤ n− 2 and i is even12i + 7, 2 ≤ i ≤ n− 2 and i is odd12n− 13, i = n− 1

f ∗(viyi) =

1, i = 111, i = 221, i = 312i− 15, 4 ≤ i ≤ n− 1 and i is even12i− 13, 4 ≤ i ≤ n− 1 and i is odd

f ∗(yivi+1) =


f ∗(uizi) =


and f ∗(zivi+1) =

5, i = 113, i = 212i− 7, 3 ≤ i ≤ n− 1 and i is odd12i− 9, 3 ≤ i ≤ n− 1 and i is even.

Thus, f is an odd sum labeling for n ≥ 5.When n = 3, an odd sum labeling of the graph is given below.


Figure 2

Hence, S(SLn) is an odd sum graph. �

Proposition 2.4. The graph S(Cp �K1) is an odd sum graph.

Proof. Let u1, v1, u2, v2, . . . , up and vp be the vertices on the cycle and uiyixi be thepath on 3 vertices attached at each ui.Case (i). p is even.

Define f : V (S(Cp �K1))→ {0, 1, 2, . . . , 4p} as follows:

f(ui) =

{4i− 3, 1 ≤ i ≤ p and i is odd4i− 1, 1 ≤ i ≤ p and i is even

f(vi) =

4i + 2, 1 ≤ i ≤ p

2− 1 and i is odd

4i, 1 ≤ i ≤ p2− 1 and i is even

4i + 4, p2≤ i ≤ p− 1 and i is odd

4i + 2, p2≤ i ≤ p− 1 and i is even

0, i = p

f(xi) =

{4i− 1, 1 ≤ i ≤ p and i is odd4i− 3, 1 ≤ i ≤ p and i is even

f(yi) =

4i− 2, 1 ≤ i ≤ p

2and i is odd

4i− 4, 1 ≤ i ≤ p2

and i is even4i, p

2+ 1 ≤ i ≤ p− 1 and i is odd

4i− 2, p2

+ 1 ≤ i ≤ p− 1 and i is even0, i = p.


f ∗(uivi) =

8i− 1, 1 ≤ i ≤ p2− 1

8i + 1, p2≤ i ≤ p− 1

4p− 1, i = p


f ∗(viui+1) =

8i + 5, 1 ≤ i ≤ p

2− 1 and i is odd

8i + 1, 1 ≤ i ≤ p2− 1 and i is even

8i + 7, p2≤ i ≤ p− 1 and i is odd

8i + 3, p2≤ i ≤ p− 1 and i is even

f ∗(uiyi) =

{8i− 5, 1 ≤ i ≤ p

28i− 3, p

2+ 1 ≤ i ≤ p

f ∗(yixi) =

8i− 3, 1 ≤ i ≤ p

2and i is odd

8i− 7, 1 ≤ i ≤ p2

and i is even8i− 1, p

2+ 1 ≤ i ≤ p and i is odd

8i− 5, p2

+ 1 ≤ i ≤ p and i is even

and f ∗(vpu1) = 1.

Thus, f is an odd sum labeling of S(Cp �K1).Case (ii). p is odd, p ≡ 1(mod 4).

The labeling f : V (S(Cp �K1))→ {0, 1, 2, . . . , 4p} is defined as follows:

f(ui) =

4i− 2, 1 ≤ i ≤ p and i is even4i− 4, 1 ≤ i ≤ p−3

2and i is odd

4i, p−12≤ i ≤ p− 1 and i is odd

4p− 2, i = p

f(xi) =

4i− 4, 1 ≤ i ≤ p−12

and i is even4i, p+3

2≤ i ≤ p and i is even

4i− 2, 1 ≤ i ≤ p−32

and i is odd4i− 4, i = p+1

2

4i− 2, p+32≤ i ≤ p− 1 and i is odd

4p, i = p

f(yi) =

4i− 5, 1 ≤ i ≤ p−1

2and i is even

4i− 3, p+32≤ i ≤ p and i is even

4i− 3, 1 ≤ i ≤ p+12

and i is odd4i− 5, p+3

2≤ i ≤ p− 1 and i is odd

4p− 1, i = p

and f(vi) =

4i + 1, 1 ≤ i ≤ p−3

2and i is odd

4i− 1, p+12≤ i ≤ p− 2 and i is odd

4i− 1, 1 ≤ i ≤ p−12

and i is even4i + 1, p+3

2≤ i ≤ p− 1 and i is even

4p− 5, i = p.


f ∗(uivi) =

8i− 3, 1 ≤ i ≤ p−12

8i− 1, p+12≤ i ≤ p− 1

8p− 7, i = p


f ∗(viui+1) =

8i + 3, 1 ≤ i ≤ p−32

and i is odd8i + 1, p+1

2≤ i ≤ p− 2 and i is odd

8i− 1, 1 ≤ i ≤ p−52

and i is even8i + 3, i = p−1

2

8i + 5, p+32≤ i ≤ p− 3 and i is even

8p− 5, i = p− 1

f ∗(vpu1) = 4p− 5

f ∗(uiyi) =

8i− 7, 1 ≤ i ≤ p−1

2

8i− 3, i = p+12

8i− 5, p+32≤ i ≤ p− 1

8p− 3, i = p

and f ∗(yixi) =

8i− 5, 1 ≤ i ≤ p−3

2and i is odd

8i− 7, p+12≤ i ≤ p− 2 and i is odd

8p− 1, i = p8i− 9, 1 ≤ i ≤ p−1

2and i is even

8i− 3, p+32≤ i ≤ p− 1 and i is even.

Thus, f is an odd sum labeling of S(Cp �K1).Case (iii). p is odd and p ≡ 3(mod 4).

The labeling f is defined as follows:

f(ui) =

4i− 4, 1 ≤ i ≤ p−1

2and i is odd

4i− 2, p+32≤ i ≤ p and i is odd

4i− 2, 1 ≤ i ≤ p−32

and i is even4i, p+1


f(vi) =

4i + 1, 1 ≤ i ≤ p− 2 and i is odd4p− 3, i = p4i + 3, 1 ≤ i ≤ p− 1 and i is even

f(xi) =

4i− 2, 1 ≤ i ≤ p−1

2and i is odd

4i, p+32≤ i ≤ p and i is odd

4i− 4, 1 ≤ i ≤ p+12

and i is even4i− 2, p+5


and f(yi) =

4i− 3, 1 ≤ i ≤ p− 2 and i is odd4p− 1, i = p4i− 5, 1 ≤ i ≤ p− 1 and i is even.


f ∗(uivi) =

8i− 3, 1 ≤ i ≤ p−12

8i− 1, p+12≤ i ≤ p− 1

8p− 5, i = p


f ∗(viui+1) =

8i + 3, 1 ≤ i ≤ p−5

2and i is odd

8i + 5, p−12≤ i ≤ p− 2 and i is odd

8i− 1, 1 ≤ i ≤ p−32

and i is even8i + 1, p+1

2≤ i ≤ p− 1 and i is even

f ∗(vpu1) = 4p− 3

f ∗(uiyi) =

8i− 7, 1 ≤ i ≤ p−12

8i− 5, p+12≤ i ≤ p− 1

8p− 3, i = p

and f ∗(yixi) =

8i− 5, 1 ≤ i ≤ p−1

2and i is odd

8i− 3, p+32≤ i ≤ p− 1 and i is odd

8p− 1, i = p8i− 9, 1 ≤ i ≤ p+1

2and i is even

8i− 7, p+52≤ i ≤ p− 1 and i is even.

Hence, f is an odd sum labeling of S(Cp �K1). �

Proposition 2.5. The graph S(Hn �K1) is an odd sum graph.

Proof. Let u1, u2, . . . , un and v1, v2, . . . , vn be the vertices of the paths of length n−1.Let a1,ia2,iui be the path attached at each ui, 1 ≤ i ≤ n and b1,ib2,ivi be the pathattached at each vi, 1 ≤ i ≤ n. Each edge uiui+1 is subdivided by a vertex xi, 1 ≤i ≤ n− 1 and each edge vivi+1 is subdivided by a vertex yi, 1 ≤ i ≤ n− 1. The edgeun+1

2vn+1

2is divided by a vertex z when n is odd. The edge un+2

2vn

2is divided by a

vertex z when n is even.Case (i). n is odd.

The labeling f : V (S(Hn �K1))→ {0, 1, 2, . . . , 8n− 2} is defined as follows:

f(ui) =


f(vi) =

{4(n + i)− 4, 1 ≤ i ≤ n and i is odd4(n + i)− 2, 1 ≤ i ≤ n and i is even

f(a1,i) =


f(a2,i) =

1, i = 14i− 5, 2 ≤ i ≤ n and i is odd4i− 3, 1 ≤ i ≤ n and i is even

f(b1,i) =



f(b2,i) =

4(n + i)− 5, 1 ≤ i ≤ n−1

2and i is odd

4(n + i)− 3, n+12≤ i ≤ n and i is odd

4(n + i)− 7, 1 ≤ i ≤ n−12

and i is even4(n + i)− 5, n+1

2≤ i ≤ n and i is even

f(xi) =


f(yi) =

4(n + i)− 1, 1 ≤ i ≤ n−1

2and i is odd

4(n + i) + 1, n+12≤ i ≤ n− 1 and i is odd

4(n + i)− 3, 1 ≤ i ≤ n−12


2≤ i ≤ n− 1 and i is even

and f(z) = 6n− 3.


f ∗(uixi) = 8i− 3, 1 ≤ i ≤ n− 1

f ∗(xiui+1) =


f ∗(viyi) =

{8(n + i)− 5, 1 ≤ i ≤ n−1

28(n + i)− 3, n+1

2≤ i ≤ n− 1

f ∗(yivi+1) =

8(n + i) + 1, 1 ≤ i ≤ n−1

2and i is odd

8(n + i) + 3, n+12≤ i ≤ n− 1 and i is odd

8(n + i)− 3, 1 ≤ i ≤ n−12



f ∗(a1,ia2,i) =


f ∗(a2,iui) =

{3, i = 18i− 7, 2 ≤ i ≤ n

f ∗(b2,ib1,i) =

8(n + i)− 7, 1 ≤ i ≤ n−1

2and i is odd

8(n + i)− 5, n+12≤ i ≤ n and i is odd

8(n + i)− 11, 1 ≤ i ≤ n−12



f ∗(vib2,i) =

{8(n + i)− 9, 1 ≤ i ≤ n−1

28(n + i)− 7, n+1

2≤ i ≤ n

f ∗(un+12z) = 8n− 3

and f ∗(zvn+12

) = 12n− 5.


Case (ii). n is even.The labeling f : V (S(Hn �K1))→ {0, 1, 2, . . . , 8n− 2} is defined as follows:

f(ui) =


f(a1,i) =


f(a2,i) =


f(vi) =


f(b1,i) =


f(b2,i) =

4(n + i)− 5, i = 14(n + i)− 7, 2 ≤ i ≤ n

2and i is odd

4(n + i)− 5, n+22≤ i ≤ n and i is odd

4(n + i)− 5, 2 ≤ i ≤ n−22

and i is even4(n + i)− 3, n


f(xi) =

{4i− 1, 1 ≤ i ≤ n− 1 and i is odd4i + 1, 1 ≤ i ≤ n and i is even

f(yi) =

4(n + i)− 3, 1 ≤ i ≤ n−2

2and i is odd

4(n + i)− 1, n2≤ i ≤ n− 1 and i is odd

4(n + i)− 1, 1 ≤ i ≤ n−22

and i is even4(n + i) + 1, n


and f(z) =

{6n− 3, n ≡ 2(mod 4)6n− 5, n ≡ 0(mod 4).


f ∗(uixi) = 8i− 3, 1 ≤ i ≤ n− 1

f ∗(xiui+1) =


f ∗(viyi) =

{8(n + i)− 5, 1 ≤ i ≤ n−2

28(n + i)− 3, n

2≤ i ≤ n− 1

f ∗(yivi+1) =

8(n + i)− 3, 1 ≤ i ≤ n−2

2and i is odd

8(n + i)− 1, n2≤ i ≤ n− 1 and i is odd

8(n + i) + 1, 1 ≤ i ≤ n−22

and i is even8(n + i) + 3, n



f ∗(a1,ia2,i) =


f ∗(a2,iui) =

{3, i = 18i− 7, 2 ≤ i ≤ n

f ∗(b2,ib1,i) =

8(n + i)− 9, i = 18(n + i)− 11, 2 ≤ i ≤ n

2and i is odd

8(n + i)− 9, n+22≤ i ≤ n and i is odd

8(n + i)− 7, 2 ≤ i ≤ n−22



f ∗(vib2,i) =

8(n + i)− 7, i = 18(n + i)− 9, 2 ≤ i ≤ n

2and i is odd

8(n + i)− 7, n+22≤ i ≤ n and i is odd

8(n + i)− 9, 2 ≤ i ≤ n−22



f ∗(un+22z) = 8n− 3

and f ∗(zvn2) =

{12n− 5, n ≡ 2(mod 4)12n− 9, n ≡ 0(mod 4).

Hence, f is an odd sum labeling. �

Proposition 2.6. The graph S(Cm@Cn) is an odd sum graph for any positive integersm,n ≥ 3.

Proof. In S(Cm@Cn), 2(m + n − 2) vertices lies on the circle and one vertex lies ona chord. Let v1, v2, . . . , v2(m+n−2) be the vertices on the cycle in S(Cm@Cn) andv2(m+n−2)+1 be the vertex having neighbours v2n−2 and v2(m+n−2).Case (i). m and n are odd with n ≥ m.

f(vi) =

2m + 2n− 4, i = 1i− 1, 2 ≤ i ≤ m + 2n− 4 and i is eveni + 1, m + 2n− 2 ≤ i ≤ 2m + 2n− 4 and i is eveni− 3, 3 ≤ i ≤ m + n− 1 and i is oddi− 1, m + n− 3 ≤ i ≤ 2m + 2n− 4 and i is odd2m + 2n− 2, i = 2m + 2n− 3.

The induced edge label is obtained as follows:

f ∗(vivi+1) =

2m + 2n− 3, i = 12i− 3, 2 ≤ i ≤ m + n− 12i− 1, m + n ≤ i ≤ m + 2n− 32i + 1, m + 2n− 2 ≤ i ≤ 2m + 2n− 5

f ∗(v2n−2v2m+2n−3) = 2m + 4n− 5 and

f ∗(v2m+2n−4v2m+2n−3) = 4m + 4n− 5.


Case (ii). m is odd, n is even and n ≥ m + 3.

f(vi) =

2m + 2n− 4, i = 1i− 1, 2 ≤ i ≤ m + n− 1 and i is eveni + 1, m + n + 1 ≤ i ≤ 2m + 2n− 4 and i is eveni− 3, 3 ≤ i ≤ m + 2n− 2 and i is oddi− 1, m + 2n− 4 ≤ i ≤ 2m + 2n− 4 and i is even2m + 2n− 2, i = 2m + 2n− 3


f ∗(vivi+1) =

2m + 2n− 3, i = 12i− 3, 2 ≤ i ≤ m + n− 12i− 1, m + n ≤ i ≤ m + 2n− 12i + 1, m + 2n ≤ i ≤ 2m + 2n− 5

f ∗(v2n−2v2m+2n−3) = 2m + 4n− 3 and

f ∗(v2m+2n−4v2m+2n−3) = 4m + 4n− 5

Case (iii). m is odd n = m + 1.

f(vi) =

i, 1 ≤ i ≤ 2m + 2n− 3 and i is oddi− 2, 2 ≤ i ≤ 2n− 2 and i is eveni, 2n ≤ i ≤ m + 2n− 5 and i is eveni + 2, m + 2n− 4 ≤ i ≤ 2m + 2n− 4 and i is even.


f ∗(vivi+1) =

2i− 1, 1 ≤ i ≤ 2n− 22i + 1, 2n− 1 ≤ i ≤ m + 2n− 52i + 3, m + 2n− 4 ≤ i ≤ 2m + 2n− 5

f ∗(v2m+2n−4v1) = 2m + 2n− 1

f ∗(v2n−2v2m+2n−3) = 2m + 4n− 7 and

f ∗(v2m+2n−4v2m+2n−3) = 4m + 4n− 5.

Case (iv). m is even and n ≥ m + 1 is even.

f(vi) =

i− 1, 1 ≤ i ≤ m + n− 2 and i is oddi + 1, m + n ≤ i ≤ 2m + 2n− 3 and i is oddi− 1, 2 ≤ i ≤ m + 2n− 4 and i is eveni + 1, m + 2n− 5 ≤ i ≤ 2m + 2n− 4 and i is even.


f ∗(vivi+1) =

2i− 1, 1 ≤ i ≤ m + n− 22i + 1, m + n− 1 ≤ i ≤ m + 2n− 42i + 3, m + 2n− 3 ≤ i ≤ 2m + 2n− 4


f ∗(v2m+2n−4v1) = 2m + 2n− 3 and

f ∗(v2n−2v2m+2n−3) = 2m + 4n− 5.

Case (v). m is even and n ≥ m + 2 is even.

f(vi) =

i− 1, 1 ≤ i ≤ m + 2n− 3 and i is oddi + 1, m + 2n− 4 ≤ i ≤ 2m + 2n− 3 and i is oddi− 1, 2 ≤ i ≤ m + n− 2 and i is eveni + 1, m + n− 1 ≤ i ≤ 2m + 2n− 4 and i is even.


f ∗(vivi+1) =

2i− 1, 1 ≤ i ≤ m + n− 22i + 1, m + n− 1 ≤ i ≤ m + 2n− 32i + 3, m + 2n− 2 ≤ i ≤ 2m + 2n− 4

f ∗(v2m+2n−4v1) = 2m + 2n− 3 and

f ∗(v2n−2v2m+2n−3) = 2m + 4n− 3.

Case (vi). m is even and n = m.

f(vi) =

i− 1, 1 ≤ i ≤ m− 1 and i is oddi + 1, m ≤ i ≤ 2m− 3 and i is odd6m− i− 3, 2m− 2 ≤ i ≤ 4m− 3 and i is oddi− 1, 2 ≤ i ≤ 2m− 2 and i is even6m− i− 3, 2m− 1 ≤ i ≤ 3m− 2 and i is even6m− i− 5, 3m− 1 ≤ i ≤ 4m− 4 and i is even.


f ∗(vivi+1) =

2i− 1, 1 ≤ i ≤ m− 12i + 1, m ≤ i ≤ 2m− 36m− 5, i = 2m− 212m− 2i− 7, 2m− 1 ≤ i ≤ 3m− 212m− 2i− 9, 3m− 1 ≤ i ≤ 4m− 4

f ∗(v4m−4v1) = 2m− 1 and

f ∗(v2m−2v4m−3) = 4m− 3.

Hence, f is an odd sum labeling. Hence, S(Cm@Cn) is an odd sum graph. �

Proposition 2.7. The Grid S(Pm × Pn) is an odd sum graph.

Proof. Let ui,j, 1 ≤ i ≤ m and 1 ≤ j ≤ n be the vertices of the grid Pm × Pn. Let vi,jbe the vertex divides the edge ui,jui,j+1 for each 1 ≤ i ≤ m and 1 ≤ j ≤ n − 1 andwi,j be the vertex divides the edge ui,jui+1,j for each 1 ≤ i ≤ m− 1 and 1 ≤ j ≤ n.


We define f : V (S(Pm × Pn))→ {0, 1, 2, . . . , 4mn− 2m− 2n} as follows:

f(ui,j) =

{2(i− 1)(2n− 1) + 2j − 2, 1 ≤ i ≤ m and i is odd, 1 ≤ j ≤ n2(i− 2)(2n− 1) + 6n− 2j − 2, 1 ≤ i ≤ m and i is even, 1 ≤ j ≤ n

f(vi,j) =

2j − 1, i = 1, 1 ≤ j ≤ n− 1(2i− 3)(2n− 1) + 4j − 2, 3 ≤ i ≤ m and i is odd, 1 ≤ j ≤ n− 1(2i− 1)(2n− 1)− 4j, 2 ≤ i ≤ m and i is even, 1 ≤ j ≤ n− 1

f(wi,j) =

{(2i + 1)(2n− 1)− 4j + 2, 1 ≤ i ≤ m− 1 and i is odd, 1 ≤ j ≤ n(2i− 1)(2n− 1) + 4(j − 1), 1 ≤ i ≤ m− 1 and i is even, 1 ≤ j ≤ n.


f ∗(ui,jvi,j) =

4j − 3, i = 1 and 1 ≤ j ≤ n− 1(4i− 5)(2n− 1) + 6j − 4, 3 ≤ i ≤ m and i is odd,

1 ≤ j ≤ n− 1(4i− 5)(2n− 1) + 6n− 6j − 2, 2 ≤ i ≤ m and i is even,

1 ≤ j ≤ n− 1

f ∗(vi,jui,j+1) =

4j − 1, i = 1 and 1 ≤ j ≤ n− 1(4i− 5)(2n− 1) + 6j − 2, 3 ≤ i ≤ m and i is odd,

1 ≤ j ≤ n− 1(4i− 5)(2n− 1) + 6n− 6j − 4, 2 ≤ i ≤ m and i is even,

1 ≤ j ≤ n− 1

f ∗(ui,jwi,j) =

(4i− 1)(2n− 1)− 2j, 1 ≤ i ≤ m− 1 and i is odd,

1 ≤ j ≤ n(4i− 5)(2n− 1) + 6n + 2j − 6, 1 ≤ i ≤ m− 1 and i is even,

1 ≤ j ≤ n

and

f ∗(wi,jui+1,j) =

(4i− 1)(2n− 1) + 6n− 6j, 1 ≤ i ≤ m− 1 and i is odd,

1 ≤ j ≤ n(4i− 1)(2n− 1) + 6j − 6, 1 ≤ i ≤ m− 1 and i is even,

1 ≤ j ≤ n

Hence, f is an odd sum labeling. The odd sum labeling is shown in the followingFigure 3. �Proposition 2.8. Let G be a graph obtained by duplicating a vertex in Pn. ThenS(G) is an odd sum graph.

Proof. Let v1, v2, . . . , vn be the vertices of Pn.Case (i). v′1 or v′n is the duplicating vertex of v1 (or vn) in Pn. Let ui,1 ≤ i ≤ n − 1 be the subdividing vertices of the edge vivi+1 and u′1 be the sub-dividing vertex of the edge v′1v2.


Figure 3. S(P5 × P8).

Subcase (i). n is odd.The vertex labeling f : V (S(G))→ {0, 1, 2, . . . , 2n} is defined as follows:

f(v′1) = 0, f(u′1) = 1, f(v1) = 4, f(v2) = 2, f(v3) = 6

f(vi) =

{2i− 2, 4 ≤ i ≤ n and i is odd2i + 2, 4 ≤ i ≤ n and i is even

f(u1) = 3, f(u2) = 7, f(u3) = 5, f(u4) = 9 and

f(ui) =

{2i + 3, 5 ≤ i ≤ n− 1 and i is odd2i− 1, 5 ≤ i ≤ n− 1 and i is even.

The induced edge labeling f ∗ is obtained as follows:

f ∗(v′1u′1) = 1, f ∗(u′1v2) = 3

f ∗(viui) =

2i + 5, 1 ≤ i ≤ 34i + 3, i = 44i + 1, 5 ≤ i ≤ n− 1

and

f ∗(uivi+1) =

5, i = 12i + 3, 2 ≤ i ≤ 44i + 7, 5 ≤ i ≤ n− 1 and i is odd4i− 1, 5 ≤ i ≤ n− 1 and i is even.

Thus, f is an odd sum labeling.


Subcase (ii). n is even.The vertex labeling f : V (S(G))→ {0, 1, 2, . . . , 2n} is defined as follows:

f(v′1) = 0, f(u′1) = 1, f(v1) = 3, f(v2) = 2,

f(vi) =


f(u1) = 4 and

f(ui) =

{2i + 3, 2 ≤ i ≤ n− 1 and i is even2i− 1, 2 ≤ i ≤ n− 1 and i is odd.


f ∗(v′1u′1) = 1,

f ∗(u′1v2) = 3,

f ∗(viui) =

{7, i = 12i + 5, 2 ≤ i ≤ n− 1

and f ∗(uivi+1) =

5, i = 14i + 7, 2 ≤ i ≤ n− 1 and i is even4i− 1, 3 ≤ i ≤ n− 1 and i is odd.

Thus, f is an odd sum labeling.Case (ii). v′i is the duplicating vertex of vi in Pn, 2 ≤ i ≤ n− 1.

Let ui, 1 ≤ i ≤ n− 1 be the subdividing vertex of the edge vivi+1 and u′i, u′′i be the

subdividing vertices of the edges vi−1v′i and v′ivi+1 respectively. The vertex labeling

f : V (S(G))→ {0, 1, 2, . . . , 2n + 2} is defined as follows:

f(vj) =

{2j − 1, 1 ≤ j ≤ i− 12j + 1, i ≤ j ≤ n

f(uj) =

{2j − 2, 1 ≤ j ≤ i− 22j + 4, i− 1 ≤ j ≤ n− 1

f(v′i) = 2i− 1, f(u′i) = 2i− 4, and f(u′′i ) = 2i− 2.


f ∗(vjuj) =

4j − 3, 1 ≤ j ≤ i− 24i− 1, j = i− 14j + 5, 1 ≤ j ≤ n− 1

f ∗(ujvj+1) =

{4j − 1, 1 ≤ j ≤ i− 24j + 7, i− 1 ≤ j ≤ n− 1

f ∗(vi−1u′i) = 4i− 7, f ∗(u′iv

′i) = 4i− 5, f ∗(v′iu

′′i ) = 4i− 3, and f ∗(u′′i vi+1) = 4i + 1.

Thus, f is an odd sum labeling. �

Proposition 2.9. Let G be a graph obtained by duplicating a vertex in Cn. ThenS(G) is an odd sum graph.


Proof. Let v1, v2, . . . , vn be the vertices of Cn and let v′2 be the duplicating vertex ofv2 in Cn. Let ui, 1 ≤ i ≤ n − 1 and let un be the subdividing vertices of the edgesvivi+1 and vnv1 respectively and u′2, u

′′2 be the subdividing vertices of the edges v1v

′2

and v′2v3 respectively.Case (i). n is odd and n ≥ 7.

The vertex labeling f : V (S(G))→ {0, 1, 2, . . . , 2n + 4} is defined as follows:

f(v1) = 2

f(vi) =

{2i + 2, 2 ≤ i ≤ n−1

22i + 4, n+1

2≤ i ≤ n

f(ui) =

{2i + 1, 1 ≤ i ≤ 22i + 3, 3 ≤ i ≤ n

f(v′2) = 0, f(u′2) = 1, and f(u′′2) = 7.


f ∗(viui) =

5, i = 111, i = 24i + 5, 3 ≤ i ≤ n−1

24i + 7, n+1

2≤ i ≤ n

f ∗(uivi+1) =

4i + 5, 1 ≤ i ≤ 24i + 7, 3 ≤ i ≤ n−3

24i + 9, n−1

2≤ i ≤ n− 1

f ∗(unv1) = 2n + 5, f ∗(v1u′2) = 3, f ∗(u′2v

′2) = 1, f ∗(v′2u

′′2) = 7

and f ∗(u′′2v3) = 15.

Thus, f is an odd sum labeling.Case (ii). n is even and n ≥ 6.

The vertex labeling f : V (S(G))→ {0, 1, 2, . . . , 2n + 4} is defined as follows:

f(vi) =

{2i + 1, 1 ≤ i ≤ n

22i + 3, n

2+ 1 ≤ i ≤ n

f(ui) = 2i + 4, 1 ≤ i ≤ n

f(v′2) = 1, f(u′2) = 2 and f(u′′2) = 0.


f ∗(viui) =

{4i + 5, 1 ≤ i ≤ n

24i + 7, n

2+ 1 ≤ i ≤ n

f ∗(uivi+1) =

{4i + 7, 1 ≤ i ≤ n

2− 1

4i + 9, n2≤ i ≤ n− 1

f ∗(unv1) = 2n + 7, f ∗(v1u′2) = 5, f ∗(u′2v

′2) = 3, f ∗(v′2u

′′2) = 1

and f ∗(u′′2v3) = 7.


Thus, f is an odd sum labeling.The odd sum labeling for G when n = 3, 4, 5 are given as follows in the Figure 4.

Figure 4

�

References

[1] F. Buckley and F. Harary, Distance in graphs, Addison-Wesley, Reading, 1990.

[2] R. Balakrishnan, A. Selvam and VYegnanarayanan, On felicitous labelings of graphs, GraphTheory and its Applications, (1996), 47–61.

[3] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 17(2011), # DS6.

[4] K. Manickam and M. Marudai, Odd mean labelings of graphs, Bulletin of Pure and AppliedSciences, 25E(1) (2006), 149–153.

[5] R. Ponraj, J. Vijaya Xavier Parthipan and R. Kala, Some results on pair sum labeling of graphs,International Journal of Mathematical Combinatorics, 4 (2010), 53–61.

[6] S. Somasundaram and R. Ponraj, Mean labelings of graphs, National Academy Science Letter,26 (2003), 210–213.

[7] S. Avadayappan and R. Vasuki, Some results on mean graphs, Ultra Scientist of Physical Sci-ences, 21(1)M (2009), 273–284.

1Department of Mathematics,Mepco Schlenk Engineering College,Sivakasi-626 005, Tamilnadu, INDIA.E-mail address: [email protected]

2Department of Mathematics,Kamaraj College of Engineering and Technology,Virudhunagar- 626 001, Tamilnadu, INDIA.E-mail address: [email protected]

3Department of Mathematics,M.D.T. Hindu College,Tirunelveli-627 010, Tamilnadu, INDIA.E-mail address: [email protected]

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ODD SUM LABELING OF SOME SUBDIVISION GRAPHS 1