On Super (a,d) - Edge Antimagic Total Labeling of Union of Two …joics.org/gallery/ics-1524.pdf · 2019-11-02 · 3 super (a,d)-edge antimagic total labeling of disconnected graphs

On Super (a, d) - Edge Antimagic Total Labeling

of Union of Two Different Sets of Stars

C. Palanivelu and N.Neela

Department of Mathematics, Knowledge Institute of Technology, Salem,

Tamil Nadu, India.

e-mail: [email protected]; [email protected]

Abstract

An (a, d) -edge antimagic total labeling of a (p, q) -graph G is a bijection

f : V ∪ E → {1, 2, 3, · · · , p + q} with the property that the edge-weight

w(uv) = f(u) + f(v) + f(uv), uv ∈ E(G), form an arithmetic progression

a, a+d, · · · , a+(q−1)d , where a > 0 and d ≥ 0 are two fixed integers. If G

admits such a labeling , then G is called an (a, d) -edge antimagic total graph.

Further, if the vertex labels are distinct integers from {1, 2, 3, · · · , p} , then f

is called a super (a, d) -edge antimagic total labeling of G (in short(a, d) -

SEAMT labeling) and a graph which admits such labeling is called super

(a, d) -edge antimagic total graph (in short(a, d) -SEAMT graph). If d = 0 ,

then the graph G is called super edge-magic total graph. In this paper,

we investigate the existence of super (a, d) -edge antimagic total labeling of

nK1,r∪mK1,s for odd n ≥ 3 , even m ≥ 2 , r, s ≥ 3 , d = 2 and δ(m,n) = 5 ,

where δ(m,n) denotes the difference between m and n .

Keywords: Magic labelling and antimagic labelling.

Mathematics Subject Classification (2000): 05C78.

1

Journal of Information and Computational Science

Volume 9 Issue 11 - 2019

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1 Introduction

All graphs considered here are finite, undirected and simple. A (p, q) -graph is a

graph G such that |V (G)| = p and |E(G)| = q . Labeling of a graph G is a

mapping that sends some set of graph elements to a set of non-negative integers. If

the domain is the vertex / edge set of G, the labeling is called vertex / edge labeling

of G . Moreover, if the domain is V (G) ∪ E(G) then the labeling is called total

labeling.

If f is a vertex labeling of a graph G , then the weight of the edge uv ∈ E(G)

is defined as w(uv) = f(u) + f(v) . If f is a total labeling , then the weight of the

edge uv ∈ E(G) is defined as w(uv) = f(u) + f(v) + f(uv) .

By an (a, d) -edge antimagic vertex labeling of a (p, q) -graph G , we mean a

bijective function f : V (G) → {1, 2, 3, · · · , p} such that {w(uv) : uv ∈ E(G)} form

an arithmetic progression a, a+ d, a+2d, · · · , a+(q− 1)d , where a > 0 and d ≥ 0

are two fixed integers.

An (a, d) -edge antimagic total labeling of a (p, q) -graph G is a bijective function

f : V (G)∪E(G) → {1, 2, · · · , p+q} with the property that {w(uv) = f(u)+f(v)+

f(uv) } , uv ∈ E(G)} form an arithmetic progression a, a + d, · · · , a + (q − 1)d ,

where a > 0 and d ≥ 0 are two fixed integers.

If G admits such a labeling , then G is said to be an (a, d) -edge antimagic total

graph. Further, f is a super (a, d) -edge antimagic total labeling of G if the vertex

labels are the distinct integers 1, 2, · · · , p . Thus a super (a, d) -edge antimagic total

graph is a graph that admits a super (a, d) -edge antimagic total labeling.

A star is a complete bipartite graph and is denoted by K1,r . The study of magic

labelings have been introduced by Simunjuntak et.al [7], a natural extension of the

concept of magic valuation, studied by Kotzig and Rosa [5] and the concept of super

edge magic labelings defined by Enomoto et.al [2]. Many authors discussed different

forms of antimagic graphs [4,6,8]. Recently, C.Palanivelu et.al [7,8] have obtained



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super (a, d) -edge antimagic total labeling of disconnected graphs and super (a, d) -

edge antimagic total labeling of union of stars. For a good collection of results on

labeling, the authors are refered to the survey by J.A. Gallian [3]. For standard

definitions and notations not defined here may be refered to D.B. West[9].

2 Main Results

In this section, we investigate the existence of super (a, d) -edge antimagic total

labeling of nK1,r ∪ mK1,s for odd n ≥ 3 , even m ≥ 2 , r, s ≥ 3 , d = 2 and

δ(m,n) = 5 , where δ(m,n) denotes the difference between m and n .

We prove the following results.

Lemma 2.1. For odd n ≥ 3 , even m ≥ 2 ,with n > m, r, s ≥ 3 there exists a

SEATL (a,2) of nK1,r ∪mK1,s , where δ(m,n) = 5 .

Proof. Let G1 = nK1,r and G2 = mK1,s , r, s ≥ 3 . Now we denote the

centre vertex of the ith copy of the stars in G1 and G2 as vi and ui and the

pendent vertices connected to vi by vij, i = 1, 2, · · · , n, j = 1, 2, · · · , r. and ui by

uij, i = 1, 2, · · · ,m, j = 1, 2, · · · , s.

For n > m , we can write (n,m) = (5 + 2t, 2t) where t ∈ Z+

Case.1 For t = 1 , we have (n,m) = (7, 2)

We define f : V (G) ∪ E(G) → {1, 2, · · · , p+ q} by

f(vi1) = i, i = 1, 2, · · · , 7

f(vi2) = 7 + i, i = 1, 2, · · · , 7

...



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f(vij) = 7(j − 1) + i, i = 1, 2, · · · , 7, j = 2, 3, · · · , r

f(ui1) = f(v7r) + i, i = 1, 2

f(ui2) = f(u2

1) + i, i = 1, 2

...

f(uij) = f(u2

j−1) + i, i = 1, 2, j = 2, 3, · · · , s

Now we continue the labeling of vi as follows

f(v5) = f(u2s) + 1

f(v6) = f(v5) + 1

f(v4) = f(v6) + 1

f(v7) = f(v4) + 2

f(ui) = f(v6) + 16− 2i, i = 1, 2

f(v1) = f(u1) + 2

f(v2) = f(v1) + 1

f(v3) = f(v2)− 2.

We define the edge labels by

f(v5v51) = f(u2

s) + 1

f(v6v61) = f(u5u

51) + 2

f(v4v41) = f(u5u

51) + 1

f(v7v71) = f(u6u

61) + 4

f(v5v52) = f(v5v

51) + 7



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f(v6v62) = f(v6v

61) + 7

f(v4v42) = f(v4v

41) + 7

f(v7v72) = f(v7v

71) + 7

...

f(v5v5j ) = f(v5v

51) + (j − 1)7, j = 2, 3, · · · , r

f(v6v6j ) = f(v6v

61) + (j − 1)7, j = 2, 3, · · · , r

f(v4v4j ) = f(v4v

41) + (j − 1)7, j = 2, 3, · · · , r

f(v7v7j ) = f(v7v

71) + (j − 1)7, j = 2, 3, · · · , r

f(uiui1) = f(v5v

51) + 23− i, i = 1, 2

f(uiui2) = f(v5v

51) + 23− i+ 2, i = 1, 2

...

f(uiuij) = f(v5v

51) + 23− i+ (j − 1)2, i = 1, 2, j = 2, 3, · · · , s

f(v1v11) = f(v4v

41) + 7

f(v2v21) = f(v1v

11) + 2

f(v3v31) = f(v2v

21)− 1

f(v1v12) = f(v1v

11) + 7

f(v2v22) = f(v2v

21) + 7

f(v3v32) = f(v3v

31) + 7

...

f(v1v1j ) = f(v1v

11) + (j − 1)7, j = 2, 3, · · · , r

f(v2v2j ) = f(v2v

21) + (j − 1)7, j = 2, 3, · · · , r

f(v3v3j ) = f(v3v

31) + (j − 1)7, j = 2, 3, · · · , r.

One can check that, the edge weights, defined by, w(viuij) = f(vi) + f(ui

j) + f(viuij)

form an arithmetic progression a, a + d, a + 2d, · · · = 14r + 4s + 16, 14r + 4s +



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18, · · · , a = 14r + 4s+ 16, d = 2 .



f(vi1) = i, i = 1, 2, · · · , 9

f(vi2) = 9 + i, i = 1, 2, · · · , 9

...

f(vij) = 9(j − 1) + i, i = 1, 2, · · · , 9, j = 1, 2, · · · , r

f(ui1) = f(v9r) + i, i = 1, 2, · · · , 4

f(ui2) = f(u4

1) + i, i = 1, 2, · · · , 4

...

f(uij) = f(u4

j−1) + i, i = 1, 2, · · · , 4, j = 2, 3, · · · , s


f(v7) = f(u4s) + 1

f(v8) = f(v7) + 1

f(v6) = f(v8) + 1

f(v9) = f(v6) + 2

f(ui) = f(v8) + 20− 2i, i = 1, 2, · · · , 4



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f(v1) = f(u1) + 2

f(v2) = f(v1) + 1

f(v3) = f(v2)− 4

f(v5) = f(v3) + 2

f(v4) = f(v3)− 2.


f(v7v71) = f(u4

s) + 1

f(v8v81) = f(u7u

71) + 2

f(v6v61) = f(u7u

71) + 1

f(v9v91) = f(u8u

81) + 4

f(v7v72) = f(v7v

71) + 9

f(v8v82) = f(v8v

81) + 9

f(v6v62) = f(v6v

61) + 9

f(v9v92) = f(v9v

91) + 9

...

f(v7v7j ) = f(v7v

71) + (j − 1)9, j = 2, 3, · · · , r

f(v8v8j ) = f(v8v

81) + (j − 1)9, j = 2, 3, · · · , r

f(v6v6j ) = f(v6v

61) + (j − 1)9, j = 2, 3, · · · , r

f(v9v9j ) = f(v8v

81) + (j − 1)9, j = 2, 3, · · · , r

f(uiui1) = f(v7v

71) + 31− i, i = 1, 2, 3, 4

f(uiui2) = f(v7v

71) + 31− i+ 4, i = 1, 2, 3, 4

...



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f(uiuij) = f(v7v

71) + 31− i+ (j − 1)4, i = 1, 2, 3, 4, j = 2, 3, · · · , s

f(v4v41) = f(v8v

81) + 1

f(v3v31) = f(v4v

41) + 1

f(v4v42) = f(v4v

41) + 9

f(v3v32) = f(v3v

31) + 9

...

f(v4v4j ) = f(v4v

41) + (j − 1)9, j = 2, 3, · · · , r

f(v3v3j ) = f(v3v

31) + (j − 1)9, j = 2, 3, · · · , r

f(v1v11) = f(v6v

61) + 9

f(v2v21) = f(v1v

11) + 2

f(v1v12) = f(v1v

11) + 9

f(v2v22) = f(v2v

21) + 9

...

f(v1v1j ) = f(v1v

11) + (j − 1)9, j = 2, 3, · · · , r

f(v2v2j ) = f(v2v

21) + (j − 1)9, j = 2, 3, · · · , r

f(v5v51) = f(v2v

21) + 1

f(v5v52) = f(v5v

51) + 9

...

f(v5v5j ) = f(v5v

51) + (j − 1)9, j = 2, 3, · · · , r.

One can check that, the edge weights defined by w(viuij) = f(vi) + f(ui

j) + f(viuij)


24, · · · , a = 18r + 8s+ 22, d = 2 .





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f(vi1) = i, i = 1, 2, · · · , 11

f(vi2) = 11 + i, i = 1, 2, · · · , 11

...

f(vij) = 11(j − 1) + i, i = 1, 2, · · · , 11, j = 1, 2, · · · , r

f(ui1) = f(v11r ) + i, i = 1, 2, · · · , 6

f(ui2) = f(u6

1) + i, i = 1, 2, · · · , 6

...

f(uij) = f(u6

j−1) + i, i = 1, 2, · · · , 6, j = 2, 3, · · · , s


f(v9) = f(u6s) + 1

f(v10) = f(v9) + 1

f(v8) = f(v10) + 1

f(v11) = f(v8) + 2

f(ui) = f(v10) + 24− 2i, i = 1, 2, · · · , 6

f(v1) = f(u1) + 2

f(v2) = f(v1) + 1

f(v3) = f(v2)− 2

f(v6) = f(v11) + 2

f(v5) = f(v6) + 2

f(v4) = f(v5) + 2

f(v7) = f(v4) + 2.



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f(v9v91) = f(u6

s) + 1

f(v10v101 ) = f(v9v

91) + 2

f(v11−(−1+4i)v11−(−1+4i)1 ) = f(v9v

91)− 3 + 4i, i = 1, 2

f(v11−(−4+4i)v11−(−4+4i)1 ) = f(v10v

101 ) + 4i, i = 1, 2

f(v6v61) = f(v10v

101 ) + 1

f(v5v51) = f(v6v

61) + 1

f(v9v92) = f(v9v

91) + 11

f(v10v102 ) = f(v10v

101 ) + 11

f(v11−(−1+4i)v11−(−1+4i)2 ) = f(v9v

91) + 8 + 4i, i = 1, 2

f(v11−(−4+4i)v11−(−4+4i)2 ) = f(v10v

101 ) + 4i+ 11, i = 1, 2

f(v6v62) = f(v10v

101 ) + 11 + 1

f(v5v52) = f(v6v

61) + 11 + 1

...

f(v9v9j ) = f(v9v

91) + (j − 1)11, j = 2, 3, · · · , r

f(v10v10j ) = f(v10v

101 ) + (j − 1)11, j = 2, 3, · · · , r

f(v11−(−1+4i)v11−(−1+4i)j ) = f(v9v

91)− 3 + (j − 1)11 + 4i, i = 1, 2, j = 2, 3, · · · , r

f(v11−(−4+4i)v11−(−4+4i)j ) = f(v10v

101 ) + 4i+ (j − 1)11, i = 1, 2, j = 2, 3, · · · , r

f(v6v6j ) = f(v10v

101 ) + 1 + (j − 1)11, j = 2, 3, · · · , r

f(v5v5j ) = f(v6v

61) + 1 + (j − 1)11, j = 2, 3, · · · , r

f(v1v11) = f(v8v

81) + 11



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f(v2v21) = f(v1v

11) + 2

f(v3v31) = f(v2v

21)− 1

f(v1v12) = f(v1v

11) + 11

f(v2v22) = f(v2v

21) + 11

f(v3v32) = f(v3v

31) + 11

...

f(v1v1j ) = f(v1v

11) + (j − 1)11, j = 2, 3, · · · , r

f(v2v2j ) = f(v2v

21) + (j − 1)11, j = 2, 3, · · · , r

f(v3v3j ) = f(v3v

31) + (j − 1)11, j = 2, 3, · · · , r

f(uiui1) = f(v9v

91) + 39− i, i = 1, 2, · · · , 6

f(uiui2) = f(v9v

91) + 39− i+ 6, i = 1, 2, · · · , 6

...

f(uiuij) = f(v9v

91) + 39− i+ (j − 1)6, i = 1, 2, · · · , 6, j = 2, 3, · · · , s.


j) + f(viuij)


30, · · · , a = 22r + 12s+ 28, d = 2 .



f(vi1) = i, i = 1, 2, · · · , 13

f(vi2) = 13 + i, i = 1, 2, · · · , 13

...

f(vij) = 13(j − 1) + i, i = 1, 2, · · · , 13, j = 2, 3, · · · , r

f(ui1) = f(v13r ) + i, i = 1, 2, · · · , 8

f(ui2) = f(u8

1) + i, i = 1, 2, · · · , 8



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...

f(uij) = f(u8

j−1) + i, i = 1, 2, · · · , 8, j = 2, 3, · · · , s


f(v13) = f(u8s) + 5

f(v12) = f(u8s) + 2

f(v13−(−2+4i)) = f(v13)− 12 + 8i, i = 1, 2

f(v13−(−1+4i)) = f(v13)− 10 + 8i, i = 1, 2

f(v13−4i) = f(v13)− 8 + 8i, i = 1, 2

f(v13−(1+4i)) = f(v13)− 6 + 8i, i = 1, 2

f(ui) = f(v12) + 28− 2i, i = 1, 2, · · · , 8

f(v1) = f(u1) + 2

f(v2) = f(v1) + 1

f(v3) = f(v2)− 4.


f(v11v111 ) = f(u8

s) + 1

f(v12v121 ) = f(u11v

111 ) + 2

f(v13−(−1+4i)v13−(−1+4i)1 ) = f(v11v

111 )− 3 + 4i, i = 1, 2

f(v13−(−4+4i)v13−(−4+4i)1 ) = f(v12v

121 ) + 4i, i = 1, 2

f(v13−(−1+4i)v13−(−1+4i)2 ) = f(v11v

112 )− 3 + 4i, i = 1, 2



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f(v13−(−4+4i)v13−(−4+4i)2 ) = f(v12v

122 ) + 4i, i = 1, 2

...

f(v13−(−1+4i)v13−(−1+4i)j ) = f(v11v

112 )− 3 + 4i, i = 1, 2

f(v13−(−4+4i)v13−(−4+4i)j ) = f(v12v

122 ) + 4i, i = 1, 2

f(v13−(1+4i)v13−(1+4i)1 ) = f(v11v

111 )− 1 + 4i, i = 1, 2

f(v13−(2+4i)v13−(2+4i)1 ) = f(v11v

111 ) + 4i, i = 1, 2

f(v13−(1+4i)v13−(1+4i)2 ) = f(v11v

111 ) + 4i+ 12, i = 1, 2

f(v13−(2+4i)v13−(2+4i)2 ) = f(v11v

111 ) + 4i+ 13, i = 1, 2

...

f(v13−(1+4i)v13−(1+4i)j ) = f(v11v

111 )− 1 + 4i+ (j − 1)13, i = 1, 2, · · · , 8, j = 2, 3, · · · , r

f(v13−(2+4i)v13−(2+4i)j ) = f(v11v

111 ) + 4i+ (j − 1)13, i = 1, 2, · · · , 8, j = 2, 3, · · · , r

f(v1v11) = f(v10v

101 ) + 13

f(v2v21) = f(v1v

11) + 2

f(v1v12) = f(v1v

11) + 13

f(v2v22) = f(v2v

21) + 13

...

f(v1v1j ) = f(v1v

11) + (j − 1)13, j = 2, 3, · · · , r

f(v2v2j ) = f(v2v

21) + (j − 1)13, j = 2, 3, · · · , r

f(v5v51) = f(v2v

21) + 1

f(v5v52) = f(v5v

51) + 1



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...

f(v5v5j ) = f(v5v

51) + (j − 1)13, j = 2, 3, · · · , r

f(uiui1) = f(v11v

111 ) + 47− i, i = 1, 2, · · · , 8

f(uiui2) = f(v11v

111 ) + 47− i+ 8, i = 1, 2, · · · , 8

...

f(uiuij) = f(v11v

111 ) + 47− i+ (j − 1)8, i = 1, 2, · · · , 8, j = 2, 3, · · · , s.


j) + f(viuij)


36, · · · , a = 26r + 16s+ 34, d = 2 .

Case.5 (n,m) = (2t+ 5, 2t) ,for t ≥ 5


f(vi1) = i, i = 1, 2, · · · , n

f(vi2) = n+ i, i = 1, 2, · · · , n

...

f(vij) = n(j − 1) + i, i = 1, 2, · · · , n, j = 2, 3, · · · , r

f(ui1) = f(vnr ) + i, i = 1, 2, · · · ,m

f(ui2) = f(um

1 ) + i, i = 1, 2, · · · ,m

...

f(uij) = f(um

j−1) + i, i = 1, 2, · · · ,m, j = 2, 3, · · · , s



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f(vn) = f(ums ) + 5

f(vn−1) = f(ums ) + 2

f(vn−(−2+4i)) = f(vn)− 12 + 8i, i = 1, 2, · · · ,⌊n

4

⌋

f(vn−(−1+4i)) = f(vn)− 10 + 8i, i = 1, 2, · · · ,⌊n

4

⌋

f(vn−4i) = f(vn)− 8 + 8i, i = 1, 2, · · · ,⌊n

4

⌋

− 1

f(vn−(1+4i)) = f(vn)− 6 + 8i, i = 1, 2, · · · ,⌊n

4

⌋

− 1

f(ui) = f(vn−1) + 2 + 2n− 2i, i = 1, 2, · · · ,m

When m ≡ 2(mod4) ≥ 10 , the labels are defined as

f(v1) = f(u1) + 2

f(v2) = f(v1) + 1

f(v3) = f(v1)− 1


f(v1) = f(u1) + 2

f(v2) = f(v1) + 2

f(v5) = f(v2) + 1.



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f(vn−2vn−21 ) = f(um

s ) + 1

f(vn−1vn−11 ) = f(vn−2v

n−21 ) + 2

f(vn−(−1+4i)vn−(−1+4i)1 ) = f(vn−2v

n−21 )− 3 + 4i, i = 1, 2, · · · ,

⌊

n+ 1

4

⌋

− 1

f(vn−(−4+4i)vn−(−4+4i)1 ) = f(vn−2v

n−21 ) + 4i, i = 1, 2, · · · ,

⌊

n+ 1

4

⌋

− 1

f(vn−(1+4i)vn−(1+4i)1 ) = f(vn−2v

n−21 )− 1 + 4i, i = 1, 2, · · · ,

⌊n

4

⌋

− 1

f(vn−(2+4i)vn−(2+4i)1 ) = f(vn−2v

n−21 ) + 4i, i = 1, 2, · · · ,

⌊n

4

⌋

− 1

f(v1v11) = f(vn−3v

n−31 ) + n

f(v2v21) = f(v1v

11) + 2

f(v1v12) = f(v1v

11) + n

f(v2v22) = f(v2v

21) + n

...

f(v1v1j ) = f(v2v

21) + (j − 1)n, j = 2, 3, · · · , r

f(v2v2j ) = f(v2v

21) + (j − 1)n, j = 2, 3, · · · , r

f(uiui1) = f(vn−2v

n−21 ) + 3n+m− i, i = 1, 2, · · · ,m

f(uiui2) = f(vn−2v

n−21 ) + 3n+m− i+m, i = 1, 2, · · · ,m

...

f(uiuij) = f(vn−2v

n−21 ) + 3n+m− i+ (j − 1)m, i = 1, 2, · · · ,m, j = 2, 3, · · · , s.


j) + f(viuij)

from an arithmetic progression a, a + d, a + 2d, · · · = 30r + 20s + 40, 30r + 20s +

42, · · · , 34r + 24s+ 46, 34r + 24s+ 48, · · · , a = 3(nr +ms) + 2m+ n− 1, d = 2 .

Lemma 2.2. For odd n ≥ 3 , even m ≥ 2 with n < m r, s ≥ 3 , there exists a

SEATL (a,2) of nK1,r ∪mK1,s and δ(m,n) = 5 .



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Proof. Let G1 = nK1,r and G2 = mK1,s with n < m , r, s ≥ 3 . Now we denote

the centre vertices of the ith copy of the stars inG1 and G2 as vi and ui and the

pendent vertices connected to vi by vij, i = 1, 2, · · · , n, j = 1, 2, · · · , r. and ui by

uij, i = 1, 2, · · · ,m, j = 1, 2, · · · , s.

Let n < m , we can write (n,m) = (2t+ 1, 2t+ 6) where t ∈ Z+



f(vi1) = i, i = 1, 2, 3

f(vi2) = 3 + i, i = 1, 2, 3

...

f(vij) = 3(j − 1) + i, i = 1, 2, 3, j = 2, 3, · · · , r

f(ui1) = f(v3r) + i, i = 1, 2, · · · , 8

f(ui2) = f(u8

1) + i, i = 1, 2, · · · , 8

...

f(uij) = f(u8

j−1) + i, i = 1, 2, · · · , 8, j = 2, 3, · · · , s

Now we continue the labeling of ui as follows

f(u6) = f(u8s) + 1

f(u7) = f(v6) + 1

f(u5) = f(u7) + 1

f(u8) = f(u5) + 2



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f(vi) = f(u7) + 8− 2i, i = 1, 2, 3

f(u1) = f(v1) + 3

f(u2) = f(u1)− 2

f(u3) = f(u1)− 4

f(u4) = f(u2) + 1.


f(vivi1) = f(u8

s) + 15− i, i = 1, 2, 3

f(vivi2) = f(viv

11) + 3, i = 1, 2, 3

...

f(vivij) = f(viv

i1) + (j − 1)8, j = 2, 3, · · · , s

f(u6u61) = f(u8

s) + 11 + 3r + 1

f(u7u71) = f(u6u

61) + 2

f(u6u62) = f(u6u

61) + 8

f(u7u72) = f(u7u

71) + 8

...

f(u6u6j) = f(u6u

61) + (j − 1)8, i = 1, 2, · · · , 8, j = 2, 3, · · · , s

f(u7u7j) = f(u7u

71) + (j − 1)8, i = 1, 2, · · · , 8, j = 2, 3, · · · , s

f(u5u51) = f(u7u

71) + 1

f(u8u81) = f(u7u

71) + 4

f(u4u41) = f(u8u

81) + 1

f(u5u52) = f(u5u

51) + 8



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f(u8u82) = f(u8u

81) + 8

f(u4u42) = f(u4u

41) + 8

...

f(u5u5j) = f(u5u

51) + 8(j − 1), j = 2, 3, · · · , s

f(u8u8j) = f(u8u

81) + 8(j − 1), j = 2, 3, · · · , s

f(u4u4j) = f(u4u

41) + 8(j − 1), j = 2, 3, · · · , s

f(u1u11) = f(u7u

71) + 8

f(u2u21) = f(u1u

11)− 1

f(u3u31) = f(u2u

21)− 1

f(u1u12) = f(u1u

11) + 8

f(u2u22) = f(u2u

21) + 8

f(u3u32) = f(u3u

31) + 8

...

f(u1u1j) = f(u1u

11) + (j − 1)8, j = 2, 3, · · · , s

f(u2u2j) = f(u2u

21) + (j − 1)8, j = 2, 3, · · · , s

f(u3u3j) = f(u3u

31) + (j − 1)8, j = 2, 3, · · · , s.

One can chech that,the edge weights defined by w(viuij) = f(vi) + f(ui

j) + f(viuij)


21, · · · , a = 6r + 16s+ 19, d = 2 .



f(vi1) = i, i = 1, 2, · · · , 5

f(vi2) = 5 + i, i = 1, 2, · · · , 5



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...

f(vij) = 5(j − 1) + i, i = 1, 2, · · · , 5, j = 2, 3, · · · , r

f(ui1) = f(v5r) + i, i = 1, 2, · · · , 10

f(ui2) = f(u10

1 ) + i, i = 1, 2, · · · , 10

...

f(uij) = f(u10

j−1) + i, i = 1, 2, 3, · · · , 10, j = 2, 3, · · · , s


f(u8) = f(u10s ) + 1

f(u9) = f(v8) + 1

f(u7) = f(u9) + 1

f(u10) = f(u7) + 2

f(u1) = f(v1) + 1

f(u2) = f(u1) + 2

f(u3) = f(u1) + 3

f(u5) = f(u10) + 2

f(u4) = f(u5) + 2

f(u6) = f(u4) + 2

f(vi) = f(u9) + 16− 2i, i = 1, 2, · · · , 5.



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f(vivi1) = f(u10

s ) + 21− i, i = 1, 2, · · · , 5

f(vivi2) = f(viv

11) + 5, i = 1, 2, · · · , 5

...

f(vivij) = f(viv

i1) + (j − 1)10,= j, 2, 3, · · · , s

f(u8u81) = f(u10

s ) + 15 + 5r + 1

f(u9u91) = f(u8u

81) + 2

f(u8u82) = f(u8u

81) + 10

f(u9u92) = f(u9u

91) + 10

...

f(u8u8j) = f(u8u

81) + (j − 1)10, i = 1, 2, · · · , 10, j = 2, 3, · · · , s

f(u9u9j) = f(u9u

91) + (j − 1)10, i = 1, 2, · · · , 10, j = 2, 3, · · · , s

f(u7u71) = f(u9u

91)− 1

f(u10u101 ) = f(u9u

91) + 4

f(u5u51) = f(u9u

91) + 1

f(u4u41) = f(u5u

51) + 1

f(u7u72) = f(u7u

71) + 10

f(u10u102 ) = f(u10u

101 ) + 10

f(u5u52) = f(u5u

51) + 10

f(u4u42) = f(u4u

41) + 10

...

f(u7u7j) = f(u7u

71) + (j − 1)10, j = 2, 3, · · · , s

f(u10u10j ) = f(u10u

101 ) + (j − 1)10, j = 2, 3, · · · , s

f(u5u5j) = f(u5u

51) + (j − 1)10, j = 2, 3, · · · , s

f(u4u4j) = f(u4u

41) + (j − 1)10, j = 2, 3, · · · , s

f(u1u11) = f(u9u

91) + 10



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f(u2u21) = f(u1u

11)− 1

f(u3u31) = f(u2u

21)− 1

f(u1u12) = f(u1u

11) + 10

f(u2u22) = f(u2u

21) + 10

f(u3u32) = f(u3u

31) + 10

...

f(u1u1j) = f(u1u

11) + (j − 1)10, j = 2, 3, · · · , s

f(u2u2j) = f(u2u

21) + (j − 1)10, j = 2, 3, · · · , s

f(u3u3j) = f(u3u

31) + (j − 1)10, j = 2, 3, · · · , s

f(u6u61) = f(u10u

101 ) + 2

f(u6u62) = f(u6u

61) + 10

...

f(u6u6j) = f(u6u

61) + (j − 1)10 j = 2, 3, · · · , s.


j) + f(viuij)


27, · · · , a = 10r + 20s+ 25, d = 2 .



f(vi1) = i, i = 1, 2, · · · , 7

f(vi2) = 7 + i, i = 1, 2, · · · , 7

...



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f(vij) = 7(j − 1) + i, i = 1, 2, · · · , 7, j = 2, 3, · · · , r

f(ui1) = f(v7r) + i, i = 1, 2, · · · , 12

f(ui2) = f(u12

1 ) + i, i = 1, 2, · · · , 12

...

f(uij) = f(u12

j−1) + i, i = 1, 2, 3, · · · , 12, j = 2, 3, · · · , s


f(u10) = f(u12s ) + 1

f(u11) = f(u10) + 1

f(u9) = f(u11) + 1

f(u12) = f(u9) + 1

f(vi) = f(u11) + 16− 2i, i = 1, 2, · · · , 7

f(u1) = f(v1) + 3

f(u2) = f(u1)− 2

f(u3) = f(u1)− 2

f(u7) = f(u3) + 2

f(u12−(5+i)) = f(u12−(4+i)) + 2, i = 1, 2

f(u8) = f(u5) + 2

f(u4) = f(u2) + 1.



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f(vivi1) = f(u12

s ) + 27− i, i = 1, 2, · · · , 7

f(vivi2) = f(viv

11) + 7, i = 1, 2, · · · , 7

...

f(vivij) = f(viv

i1) + (j − 1)12, j = 2, 3, · · · , s

f(u10u101 ) = f(u12

s ) + 22 + 7r

f(u11u111 ) = f(u10u

101 ) + 2

f(u10u102 ) = f(u10u

101 ) + 12

f(u11u112 ) = f(u11u

111 ) + 12

...

f(u10u10j ) = f(u10u

101 ) + (j − 1)12, i = 1, 2, · · · , 12, j = 2, 3, · · · , s

f(u11u11j ) = f(u11u

111 ) + (j − 1)12, i = 1, 2, · · · , 12, j = 2, 3, · · · , s

f(u7u71) = f(u11u

111 ) + 1

f(u7u72) = f(u7u

71) + 12

...

f(u7u7j) = f(u7u

71) + (j − 1)12, j = 2, 3, · · · , s

f(u6u61) = f(u7u

71) + 1

f(u6u62) = f(u6u

61) + 12

...

f(u6u6j) = f(u6u

61) + (j − 1)12, j = 2, 3, · · · , s

f(u1u11) = f(u6u

61) + 7

f(u2u21) = f(u1u

11)− 1

f(u3u31) = f(u2u

21)− 1

f(u1u12) = f(u1u

11) + 12

f(u2u22) = f(u2u

21) + 12

f(u3u32) = f(u3u

31) + 12

...

f(u1u1j) = f(u1u

11) + (j − 1)12, j = 2, 3, · · · , s



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f(u2u2j) = f(u2u

21) + (j − 1)12, j = 2, 3, · · · , s

f(u3u3j) = f(u3u

31) + (j − 1)12, j = 2, 3, · · · , s

f(u9u91) = f(u11u

111 )− 1

f(u12u121 ) = f(u11u

111 ) + 4

f(u9u92) = f(u9u

91) + 12

f(u12u122 ) = f(u12u

121 ) + 12

...

f(u9u9j) = f(u9u

91) + (j − 1)12, j = 2, 3, · · · , s

f(u12u12j ) = f(u12u

121 ) + (j − 1)12, j = 2, 3, · · · , s

f(u5u51) = f(u6u

61) + 1

f(u8u81) = f(u1u

11) + 1

f(u4u41) = f(u8u

81) + 1

f(u5u52) = f(u5u

51) + 12

f(u8u82) = f(u8u

81) + 12

f(u4u42) = f(u4u

41) + 12

...

f(u5u5j) = f(u5u

51) + (j − 1)12, j = 2, 3, · · · , s

f(u8u8j) = f(u8u

81) + (j − 1)12, j = 2, 3, · · · , s

f(u4u4j) = f(u4u

41) + (j − 1)12, j = 2, 3, · · · , s.


j) + f(viuij)


33, · · · , a = 14r + 24s+ 31, d = 2 .





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f(vi1) = i, i = 1, 2, · · · , 9

f(vi2) = 9 + i, i = 1, 2, · · · , 9

...

f(vij) = 9(j − 1) + i, i = 1, 2, · · · , 9, j = 2, 3, · · · , r

f(ui1) = f(v9r) + i, i = 1, 2, · · · , 14

f(ui2) = f(u14

1 ) + i, i = 1, 2, · · · , 14

...

f(uij) = f(u14

j−1) + i, i = 1, 2, 3, · · · , 14, j = 2, 3, · · · , s


f(u12) = f(u14s ) + 1

f(u13) = f(u12) + 1

f(u11) = f(u13) + 1

f(u14) = f(u11) + 1

f(vi) = f(u13) + 20− 2i, i = 1, 2, · · · , 9

f(u1) = f(v1) + 1

f(u2) = f(u1) + 2

f(u3) = f(u1) + 3

f(u9) = f(u14) + 2



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f(u14−(5+i)) = f(u14−(4+i)) + 2, i = 1, 2

f(u10) = f(u7) + 2

f(u5) = f(u10) + 2

f(u4) = f(u5) + 2

f(u6) = f(u4) + 2.


f(vivi1) = f(u14

s ) + 33− i, i = 1, 2, · · · , 9

f(vivi2) = f(viv

11) + 9, i = 1, 2, · · · , 9

...

f(vivij) = f(viv

i1) + (j − 1)14,= j, 2, 3, · · · , s

f(u12u121 ) = f(u14

s ) + 24 + 9r + 1

f(u13u131 ) = f(u12u

121 ) + 2

f(u12u122 ) = f(u12u

121 ) + 14

f(u13u132 ) = f(u13u

131 ) + 14

...

f(u12u12j ) = f(u12u

121 ) + (j − 1)14, i = 1, 2, · · · , 14, j = 2, 3, · · · , s

f(u13u13j ) = f(u13u

131 ) + (j − 1)14, i = 1, 2, · · · , 14, j = 2, 3, · · · , s



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f(u1u11) = f(u13u

131 ) + 14

f(u2u21) = f(u1u

11)− 1

f(u3u31) = f(u3u

31)− 1

f(u1u12) = f(u1u

11) + 14

f(u2u22) = f(u2u

21) + 14

f(u3u32) = f(u3u

31) + 14

...

f(u1u1j) = f(u1u

11) + (j − 1)14, j = 2, 3, · · · , s

f(u2u2j) = f(u2u

21) + (j − 1)14, j = 2, 3, · · · , s

f(u3u3j) = f(u3u

31) + (j − 1)14, j = 2, 3, · · · , s

f(u11u111 ) = f(u13u

131 )− 1

f(u14u141 ) = f(u13u

131 ) + 4

f(u9u91) = f(u13u

131 ) + 1

f(u8u81) = f(u9u

91) + 1

f(u11u112 ) = f(u11u

111 ) + 14

f(u14u142 ) = f(u14u

141 ) + 14

f(u9u92) = f(u9u

91) + 14

f(u8u82) = f(u8u

81) + 14

...



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f(u11u11j ) = f(u11u

111 ) + (j − 1)14, j = 2, 3, · · · , s

f(u14u14j ) = f(u14u

141 ) + (j − 1)14, j = 2, 3, · · · , s

f(u9u9j) = f(u9u

91) + (j − 1)14, j = 2, 3, · · · , s

f(u8u8j) = f(u8u

81) + (j − 1)14, j = 2, 3, · · · , s

f(u7u71) = f(u8u

81) + 1

f(u5u51) = f(u14u

141 ) + 2

f(u4u41) = f(u5u

51) + 1

f(u10u101 ) = f(u1u

11) + 1

f(u6u61) = f(u2u

21) + 1

f(u7u72) = f(u7u

71) + 14

f(u5u52) = f(u5u

51) + 14

f(u4u42) = f(u4u

41) + 14

f(u10u102 ) = f(u10u

101 ) + 14

f(u6u62) = f(u6u

61) + 14

...

f(u7u7j) = f(u7u

71) + (j − 1)14, j = 2, 3, · · · , s

f(u5u5j) = f(u5u

51) + (j − 1)14, j = 2, 3, · · · , s

f(u4u4j) = f(u4u

41) + (j − 1)14, j = 2, 3, · · · , s

f(u10u10j ) = f(u10u

101 ) + (j − 1)14, j = 2, 3, · · · , s

f(u6u6j) = f(u6u

61) + (j − 1)14, j = 2, 3, · · · , s.


j) + f(viuij)


39, · · · , a = 18r + 28s+ 37, d = 2 .





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f(vi1) = i, i = 1, 2, · · · , 11

f(vi2) = 11 + i, i = 1, 2, · · · , 11

...

f(vij) = 11(j − 1) + i, i = 1, 2, · · · , 11, j = 2, 3, · · · , r

f(ui1) = f(v11r ) + i, i = 1, 2, · · · , 16

f(ui2) = f(u16

1 ) + i, i = 1, 2, · · · , 16

...

f(uij) = f(u16

j−1) + i, i = 1, 2, 3, · · · , 16, j = 2, 3, · · · , s


f(u16) = f(u16s ) + 5

f(u15) = f(u16s ) + 2

f(u16−(−2+4i)) = f(u16)− 12 + 8i, i = 1, 2, 3

f(u16−(−1+4i)) = f(u16)− 10 + 8i, i = 1, 2, 3

f(u16−4i) = f(u16)− 8 + 8i, i = 1, 2, 3

f(u16−(1+4i)) = f(u16)− 6 + 8i, i = 1, 2

f(vi) = f(u15) + 24− 2i, i = 1, 2, · · · , 11

f(u1) = f(v1) + 3

f(u2) = f(u1)− 2

f(u3) = f(u1)− 2.



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f(vivi1) = f(u16

s ) + 39− i, i = 1, 2, · · · , 11

f(vivi2) = f(viv

11) + 11, i = 1, 2, · · · , 11

...

f(vivij) = f(viv

i1) + (j − 1)16,= j, 2, 3, · · · , s

f(u14u141 ) = f(u16

s ) + 28 + 11r

f(u13u131 ) = f(u9u

91) + 2

f(u14u142 ) = f(u14u

141 ) + 16

f(u13u132 ) = f(u13u

131 ) + 16

...

f(u14u14j ) = f(u14u

141 ) + (j − 1)16, i = 1, 2, · · · , 16, j = 2, 3, · · · , s

f(u13u13j ) = f(u13u

131 ) + (j − 1)16, i = 1, 2, · · · , 16, j = 2, 3, · · · , s

f(u16−(−1+4i)u16−(−1+4i)1 ) = f(u15u

151 )− 3 + 4i, i = 1, 2, 3

f(u16−(1+4i)u16−(1+4i)1 ) = f(u15u

151 )− 1 + 4i, i = 1, 2, 3

f(u16−(2+4i)u16−(2+4i)1 ) = f(u15u

151 ) + 4i, i = 1, 2, 3

f(u16−(3+4i)u16−(3+4i)1 ) = f(u15u

151 ) + 2 + 4i+ 11, i = 1, 2, 3

f(u16−(−1+4i)u16−(−1+4i)2 ) = f(u16−(−1+4i)u

16−(−1+4i)1 ) + 16, i = 1, 2, 3

f(u16−(1+4i)u16−(1+4i)2 ) = f(u16−(1+4i)u

16−(1+4i)1 ) + 16, i = 1, 2, 3

f(u16−(2+4i)u16−(2+4i)2 ) = f(u16−(2+4i)u

16−(2+4i)1 ) + 16, i = 1, 2, 3

f(u16−(3+4i)u16−(3+4i)2 ) = f(u16−(3+4i)u

16−(3+4i)1 ) + 16, i = 1, 2, 3

...

f(u16−(−1+4i)u16−(−1+4i)j ) = f(u16−(−1+4i)u

16−(−1+4i)1 ) + (j − 1)16, i = 1, 2, 3, j = 2, 3, · · · , s



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f(u16−(1+4i)u16−(1+4i)j ) = f(u16−(1+4i)u

16−(1+4i)1 ) + (j − 1)16, i = 1, 2, 3, j = 2, 3, · · · , s

f(u16−(2+4i)u16−(2+4i)j ) = f(u16−(2+4i)u

16−(2+4i)1 ) + (j − 1)16, i = 1, 2, 3, j = 2, 3, · · · , s

f(u16−(3+4i)u16−(3+4i)j ) = f(u16−(3+4i)u

16−(3+4i)1 ) + (j − 1)16, i = 1, 2, 3, j = 2, 3, · · · , s

f(u1u11) = f(u1u

11) + 1

f(u4u41) = f(u1u

11) + 2

f(u1u12) = f(u1u

11) + 16

f(u4u42) = f(u4u

41) + 16

...

f(u1u1j) = f(u1u

11) + (j − 1)16, j = 2, 3, · · · , s

f(u4u4j) = f(u4u

41) + (j − 1)16, j = 2, 3, · · · , s.


j) + f(viuij)


45, · · · , a = 22r + 32s+ 43, d = 2 .



f(vi1) = i, i = 1, 2, · · · , 13

f(vi2) = 13 + i, i = 1, 2, · · · , 13

...

f(vij) = 13(j − 1) + i, i = 1, 2, · · · , 13, j = 2, 3, · · · , r

f(ui1) = f(v13r ) + i, i = 1, 2, · · · , 18

f(ui2) = f(u18

1 ) + i, i = 1, 2, · · · , 18

...

f(uij) = f(u18

j−1) + i, i = 1, 2, 3, · · · , 18, j = 2, 3, · · · , s



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f(u18) = f(u18s ) + 5

f(u17) = f(u18s ) + 2

f(u16) = f(u18s ) + 1

f(u18−(−2+4i)) = f(u16)− 12 + 8i, i = 2, 3, 4

f(u18−(−1+4i)) = f(u16)− 10 + 8i, i = 1, 2, 3

f(u18−4i) = f(u16)− 8 + 8i, i = 1, 2, 3

f(u18−(1+4i)) = f(u16)− 6 + 8i, i = 1, 2, 3

f(vi) = f(u17) + 28− 2i, i = 1, 2, · · · , 13

f(u1) = f(v1) + 1

f(u2) = f(u1) + 2

f(u3) = f(u1) + 3.


f(vivi1) = f(u18

s ) + 45− i, i = 1, 2, · · · , 13

f(vivi2) = f(viv

11) + 13, i = 1, 2, · · · , 13

...

f(vivij) = f(viv

i1) + (j − 1)18,= j, 2, 3, · · · , s

f(u16u161 ) = f(u18

s ) + 32 + 13r

f(u17u171 ) = f(u11u

111 ) + 2

f(u16u162 ) = f(u16u

161 ) + 18



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f(u17u172 ) = f(u17u

171 ) + 18

...

f(u16u16j ) = f(u16u

161 ) + (j − 1)18, i = 1, 2, · · · , 18, j = 2, 3, · · · , s

f(u17u17j ) = f(u17u

171 ) + (j − 1)18, i = 1, 2, · · · , 18, j = 2, 3, · · · , s

f(u18−(−1+4i)u18−(−1+4i)1 ) = f(u17u

171 )− 3 + 4i, i = 1, 2, 3

f(u18−(1+4i)u18−(1+4i)1 ) = f(u17u

171 )− 1 + 4i, i = 1, 2, 3

f(u18−(2+4i)u18−(2+4i)1 ) = f(u17u

171 ) + 4i, i = 1, 2, 3

f(u18−(3+4i)u18−(3+4i)1 ) = f(u17u

171 ) + 2 + 4i+ 13, i = 1, 2, 3

f(u18−(−1+4i)u18−(−1+4i)2 ) = f(u18−(−1+4i)u

18−(−1+4i)1 ) + 18, i = 1, 2, 3

f(u18−(1+4i)u18−(1+4i)2 ) = f(u18−(1+4i)u

18−(1+4i)1 ) + 18, i = 1, 2, 3

f(u18−(2+4i)u18−(2+4i)2 ) = f(u18−(2+4i)u

18−(2+4i)1 ) + 18, i = 1, 2, 3

f(u18−(3+4i)u18−(3+4i)2 ) = f(u18−(3+4i)u

18−(3+4i)1 ) + 18, i = 1, 2, 3

...

f(u18−(−1+4i)u18−(−1+4i)j ) = f(u18−(−1+4i)u

18−(−1+4i)1 ) + (j − 1)18, i = 1, 2, 3, j = 2, 3, · · · , s

f(u18−(1+4i)u18−(1+4i)j ) = f(u18−(1+4i)u

18−(1+4i)1 ) + (j − 1)18, i = 1, 2, 3, j = 2, 3, · · · , s

f(u18−(2+4i)u18−(2+4i)j ) = f(u18−(2+4i)u

18−(2+4i)1 ) + (j − 1)18, i = 1, 2, 3, j = 2, 3, · · · , s

f(u18−(3+4i)u18−(3+4i)j ) = f(u18−(3+4i)u

18−(3+4i)1 ) + (j − 1)18, i = 1, 2, 3, j = 2, 3, · · · , s

f(vivi1) = f(u18

s ) + 45− i, i = 1, 2, · · · , 13

f(vivi2) = f(viv

11) + 13, i = 1, 2, · · · , 13

...

f(vivij) = f(viv

i1) + (j − 1)18, j = 2, 3, · · · , s



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f(u1u11) = f(u4u

41) + 1

f(u2u21) = f(u1u

11) + 2

f(u3u31) = f(u2u

21) + 2

f(u6u61) = f(u3u

31)− 1

f(u1u12) = f(u1u

11) + 18

f(u2u22) = f(u2u

21) + 18

f(u3u32) = f(u3u

31) + 18

f(u6u62) = f(u6u

61) + 18

...

f(u1u1j) = f(u1u

11) + (j − 1)18, j = 2, 3, · · · , s

f(u2u2j) = f(u2u

21) + (j − 1)18, j = 2, 3, · · · , s

f(u3u3j) = f(u3u

31) + (j − 1)18, j = 2, 3, · · · , s

f(u6u6j) = f(u6u

61) + (j − 1)18, j = 2, 3, · · · , s


j) + f(viuij)


51, · · · , a = 26r + 36s+ 49, d = 2 .

Case.7 (n,m) = (2t+ 1, 2t+ 6) , for t ≥ 7


f(vi1) = i, i = 1, 2, · · · , n

f(vi2) = n+ i, i = 1, 2, · · · , n



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...

f(vij) = (j − 1)n+ i, i = 1, 2, · · · , n, j = 2, 3, · · · , r

f(ui1) = f(vnr ) + i, i = 1, 2, · · · ,m

f(ui2) = f(um

1 ) + i, i = 1, 2, · · · ,m

...

f(uij) = f(um

j−1) + i, i = 1, 2, · · · ,m, j = 2, 3, · · · , s.


f(um) = f(ums ) + 5

f(um−1) = f(ums ) + 2

f(um−(−2+4i)) = f(um)− 12 + 8i, i = 1, 2, · · · ,⌊m

4

⌋

f(um−(−1+4i)) = f(um)− 10 + 8i, i = 1, 2, · · · ,⌊m

4

⌋

f(um−4i) = f(um)− 8 + 8i, i = 1, 2, · · · ,⌊m

4

⌋

f(um−(1+4i)) = f(um)− 6 + 8i, i = 1, 2, · · · ,⌊m

4

⌋

When m = 0(mod4) ≥ 16 , the labels are defined as

f(u1) = f(v1) + 3


f(u1) = f(v1) + 1

f(u2) = f(u1) + 2

f(u3) = f(u1) + 3

f(vi) = f(um−1) + 2 + 2n− 2i, i = 1, 2, · · · , n.



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f(vivi1) = f(um

s ) +m+ 2n+ 1− i, i = 1, 2, · · · , n

f(vivi2) = f(viv

11) + n, i = 1, 2, · · · , n

...

f(vivij) = f(viv

i1) + (j − 1)m, j = 2, 3, · · · , s

f(um−2um−21 ) = f(um

s ) +m+ n+ nr + 1

f(um−1um−11 ) = f(um−2u

m−21 ) + 2

f(um−2um−22 ) = f(um−2u

m−21 ) +m

f(um−1um−12 ) = f(um−1u

m−11 ) +m

...

f(um−2um−2j ) = f(um−2u

m−21 ) + (j − 1)m, i = 1, 2, · · · ,m, j = 2, 3, · · · , s

f(um−1um−1j ) = f(um−1u

m−11 ) + (j − 1)m, i = 1, 2, · · · ,m, j = 2, 3, · · · , s

f(um−(−1+4i)um−(−1+4i)1 ) = f(um−1u

m−11 )− 3 + 4i, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(1+4i)um−(1+4i)1 ) = f(um−1u

m−11 )− 1 + 4i, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(2+4i)um−(2+4i)1 ) = f(um−1u

m−11 ) + 4i, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(3+4i)um−(3+4i)1 ) = f(um−1u

m−11 ) + 2 + 4i+ n, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(−1+4i)um−(−1+4i)2 ) = f(um−(−1+4i)u

m−(−1+4i)1 ) +m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(1+4i)um−(1+4i)2 ) = f(um−(1+4i)u

m−(1+4i)1 ) +m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(2+4i)um−(2+4i)2 ) = f(um−(2+4i)u

m−(2+4i)1 ) +m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

f(um−(3+4i)um−(3+4i)2 ) = f(um−(3+4i)u

m−(3+4i)1 ) +m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1

...

f(um−(−1+4i)um−(−1+4i)j ) = f(um−(−1+4i)u

m−(−1+4i)1 ) + (j − 1)m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1, j = 2, 3, · · ·

f(um−(1+4i)um−(1+4i)j ) = f(um−(1+4i)u

m−(1+4i)1 ) + (j − 1)m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1, j = 2, 3, · · · , s

f(um−(2+4i)um−(2+4i)j ) = f(um−(2+4i)u

m−(2+4i)1 ) + (j − 1)m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1, j = 2, 3, · · · , s

f(um−(3+4i)um−(3+4i)j ) = f(um−(3+4i)u

m−(3+4i)1 ) + (j − 1)m, i = 1, 2, · · · ,

⌊m

4

⌋

− 1, j = 2, 3, · · · , s



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f(u1u11) = f(u2u

21) + 1

f(u4u41) = f(u1u

11) + 2

f(u1u12) = f(u1u

11) +m

f(u4u42) = f(u4u

41) +m

...

f(u1u1j) = f(u1u

11) + (j − 1)m, j = 2, 3, · · · , s

f(u4u4j) = f(u4u

41) + (j − 1)m, j = 2, 3, · · · , s


f(u1u11) = f(u4u

41) + 1

f(u2u21) = f(u1u

11) + 2

f(u3u31) = f(u2u

21) + 2

f(u6u61) = f(u3u

31)− 1

f(u1u12) = f(u1u

11) +m

f(u2u22) = f(u2u

21) +m

f(u3u32) = f(u3u

31) +m

f(u6u62) = f(u6u

61) +m

...

f(u1u1j) = f(u1u

11) + (j − 1)m, j = 2, 3, · · · , s

f(u2u2j) = f(u2u

21) + (j − 1)m, j = 2, 3, · · · , s

f(u3u3j) = f(u3u

31) + (j − 1)m, j = 2, 3, · · · , s

f(u6u6j) = f(u6u

61) + (j − 1)m, j = 2, 3, · · · , s


j) + f(viuij)



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57, · · · , a = 30r + 40s+ 55, d = 2 .

We can use the Lemmas 2.1 and 2.2, to prove the following theorem.

Theorem 2.1. For odd n ≥ 3 , even m ≥ 2 , and r, s ≥ 3 there exists a super (a, 2)

edge antimagic total labeling (in short(a, d) -SEAMT labeling)of nK1,r∪mK1,s and

δ(m,n) = 5 ,where δ(m,n) denotes the difference between m and n.

Proof. Follows from the Lemmas 2.1 and 2.2

References

[1] Dafik, Mirka Miller, Joe Ryan, Martin Baca, On super (a, d) -edge-antimagic

total labelings of disconnected graphs, Discrete Math. 309 (2009) 4909-4915.

[2] H. Enomoto, A.S. Lado, T. Nakamigawa, G. Ringel, Super edge-antimagic

labelings of the generalized Petersen graph P(

n, n−12

)

, Util. Math. 70 (2006)

119-127.

[3] J.A. Gallian, A Dynamic survey of Graph labeling, The Electronic Journal of

Combinatorics (2018).

[4] Himayat Ullah et al., On super (a, d) -Edge-Antimagic total labelings of special

types of crown graphs, Journal of Applied Mathematics, (2013) , Article ID

896815 6 pages.

[5] A. Kotzig, A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13

(1970) 451-461.

[6] Martin Baca et al., On super (a, 1) -edge-antimagic total labelings of regular

graphs, Discrete Math. 310 (2010) 1408-1412.



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[7] C.Palanivelu, A.Muthusamy, N.Neela, Super (a, d) -edge antimagic total

labeling of disconnected graphs II. (submitted)

[8] C.Palanivelu, N.Neela, Super (a, d) -edge antimagic total labeling of union of

stars, International Journal of Applied Engineering Research,14 (2019) 2089-

2092.

[9] R. Simanjuntak, F. Bertault, M. Miller, Two new (a, d) -antimagic graph

labelings, in: Proc. Eleventh Australasian Workshop on Combinatorial

Algorithms, 2000, 179-189.

[10] W.D. Wallis, Magic graphs, Birkhauser, Boston, Basel, Berlin, 2001.

[11] D.B. West, An Introduction to Graph Theory, Prentice-Hall, 1996.



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