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H-E-super magic decomposition of complete bipartite graphs
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Electronic Notes in Discrete Mathematics 48 (2015) 297–300 www.elsevier.com/locate/endm
H-E-super magic decomposition of complete bipartite graphs S. Stalin Kumar 1 Department of Mathematics The American College Madurai−625 002, TamilNadu, India
G. Marimuthu 2 Department of Mathematics The Madura College Madurai−625 011, TamilNadu, India
Abstract An H-magic labeling of a H-decomposable graph G is a bijection f : V (G)∪E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, v∈V (H) f (v) + f (e) is constant. The labeling f is said to be H-E-super magic if f (E(G)) = e∈E(H) {1, 2, ..., q}. In this paper, we prove that a complete bipartite graph is H-E-super magic decomposable where H ∼ = K1,n with n ≥ 1. Keywords: H-decomposable graph, H-E-super magic labeling.
1 2
Email: [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2015.05.044 1571-0653/© 2015 Elsevier B.V. All rights reserved.
298
1
S.S. Kumar, G. Marimuthu / Electronic Notes in Discrete Mathematics 48 (2015) 297–300
Introduction
In this paper we consider only finite and simple undirected bipartite graphs. The vertex and edge sets of a graph G denoted by V (G) and E(G) respectively and we let |V (G)| = p and |E(G)| = q. For graph theoretic notations, we follow [2,3]. Many kinds of labeling have been studied and an excellent survey of graph labelings can be found in [4]. Suppose G is H-decomposable. A total labeling f : V (G) ∪ E(G) → {1, 2, · · · , p + q} is called an H-magic labeling of G if there exists a positive integer k (called magic constant) such that for every copy H in the decom position, v∈V (H) f (v) + e∈E(H) f (e) = k. A graph G that admits such a labeling is called an H-magic decomposable graph. An H-magic labeling f is called a H-E-super magic labeling if f (E(G)) = {1, 2, · · · , q}. A graph that admits a H-E-super magic labeling is called a H-E-super magic decomposable graph. For further details on H-magic type labelings refer [5,6,8,11]. In [9], Subbiah and Pandimadevi introduced the notion of H-E-super magic decomposition of graphs. Yoshimi Ecawa et al. [10] studied the decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components. The notion of star-subgraph was introduced by Akiyama and Kano in [1]. A subgraph F of a graph G is called a star-subgraph if each component of F is a star. Here by a star, we mean a complete bipartite graph of the form K1,m with m ≥ 1. In [7], Marimuthu and Stalin Kumar studied the H-V -super magic decomposition of complete bipartite graphs. A subgraph F of a graph G is called a n-star-subgraph if F ∼ = K1,n with 2 ≤ n < p.
2
Main Result
In this section, we consider the graphs G ∼ = Kn,n and H ∼ = K1,n , where n ≥ 2. 2 Clearly p = 2n and q = n . Theorem 2.1 Suppose {H1 , H2 , · · · , Hn } is an n-star-decomposition of G. Then G is an n-star-E-super magic decomposable graph with magic constant {2n + n(n+1)(3n+1) }. 2 Proof. Let U = {u1 , u2 , · · · , un } and V = {v1 , v2 , · · · , vn } be the bipartition of G. Let {H1 , H2 , · · · , Hn } be an n-star decomposition of G, where each Hi is isomorphic to H with V (Hi ) = {ui , v1 , v2 , · · · , vn } and E(Hi ) = {ui v1 , ui v2 , · · · , ui vn }, for 1 ≤ i ≤ n. Define a total labeling f : V (G) ∪ E(G) → {1, 2, · · · , p + q} by f (ui ) = n2 + (2i − 1) and f (vi ) = n2 + 2i, for all i = 1, 2, · · · , n.
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S.S. Kumar, G. Marimuthu / Electronic Notes in Discrete Mathematics 48 (2015) 297–300
Case 1. n is even. Now the edge of G are given in Table 1. Table 1 The edge label of an n-star-decomposition of G if n is even. f
v1
v2
v3
...
vn − 1
vn
u1 .. .
n
n+1
3n
...
(n − 1)n
n(n)
...
...
...
...
...
...
uk .. .
n − (k − 1)
n+k
3n − (k − 1)
...
(n − 1)n − (k − 1)
n(n) − (k − 1)
...
...
...
...
...
...
un
1
2n
2n + 1
...
(n − 2)n + 1
(n − 1)n + 1
From Table 1 and from the definition of f , we obtain f (Hk ) = f (uk ) + n n n(n+1)(3n+1) for all k, 1 ≤ k ≤ n. i=1 f (vi ) + i=1 f (uk vi ) = 2n + 2 Case 2. n is odd. Now the edge of G are given in Table 2. From Table 2 and from definition Table 2 The edge label of an n-star-decomposition of G if n is odd. f
v1
v2
...
vn − 1
u1
n
n+1
...
(n − 3)n + +2( n+1 2
n+1 2
vn − 1
vn
(n − 1)n
(n − 1)n + 1
− 1)
.. .
...
...
...
...
...
...
uk
(n + 1) − k
n+k
...
(n − 3)n
(n − 2)n
(n − 1)n
+(n + 1) − k
+(n + 1) − k
+(n + 1) − k
...
...
...
.. . un
... 1
... 2n
... ...
(n − 3)n + 1
(n − 3)n +
n+1 2
+1
n(n)
of f , we obtain , 1 ≤ k ≤ n. f (Hk ) = f (uk ) + ni=1 f (vi ) + ni=1 f (uk vi ) = 2n + n(n+1)(3n+1) 2 Hence the graph G is an n-star-E-super magic decomposable graph. 2
References [1] J. Akiyama and M. Kano, Path factors of a graph, Graphs and Applications, Wiley, Newyork, (1984).
300
S.S. Kumar, G. Marimuthu / Electronic Notes in Discrete Mathematics 48 (2015) 297–300
[2] G. Chartrand, L. Lesniak, Graphs and Digraphs, 3rd Edition, Chapman and Hall, Boca Roton, London, Newyork, Washington, D.C (1996). [3] G. Chartrand and P. Zhang, Chromatic Graph Theory, Chapman and Hall, CRC, Boca Roton, (2009). [4] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin., 16 (2013), #DS6. [5] A. Guti´ errez, A. Llad´ o, Magic coverings, J. Combin. Math. Combin. Comput., 55 (2005), 43–56. [6] A. Llad´ o, J.Moragas, Cycle-magic graphs, Discrete Math., 307 (2007), 2925– 2933. [7] G. Marimuthu, S. Stalin Kumar, H-V -super magic decomposition of complete bipartite graphs, Communicated. [8] A. A. G. Ngurah, A. N. M. Salman, L. Susilowati, H-Supermagic labeling of graphs, Discrete Math., 310 (2010),1293–1300. [9] P. Subbiah, J. Pandimadevi, H-E-super magic decompositions of graphs, Electronic Journal of Graph Theory and and Applications 2(2) (2014), 115– 128. [10] Yoshimi Egawa, Masatsugu Urabe, Toshihito Fukuda and Seiichiro Nagoya, A Decomposition of Complete bipartite graphs into edge-disjoint subgraphs with star components, Discrete Math., 58 (1986), 93–95. [11] Zhihe Liang, Cycle-super magic decompositions of complete multipartite graphs, Discrete Math., 312 (2012), 3342–3348.
Electronic Notes in Discrete Mathematics 48 (2015) 297–300 www.elsevier.com/locate/endm
H-E-super magic decomposition of complete bipartite graphs S. Stalin Kumar 1 Department of Mathematics The American College Madurai−625 002, TamilNadu, India
G. Marimuthu 2 Department of Mathematics The Madura College Madurai−625 011, TamilNadu, India
Abstract An H-magic labeling of a H-decomposable graph G is a bijection f : V (G)∪E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, v∈V (H) f (v) + f (e) is constant. The labeling f is said to be H-E-super magic if f (E(G)) = e∈E(H) {1, 2, ..., q}. In this paper, we prove that a complete bipartite graph is H-E-super magic decomposable where H ∼ = K1,n with n ≥ 1. Keywords: H-decomposable graph, H-E-super magic labeling.
1 2
Email: [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2015.05.044 1571-0653/© 2015 Elsevier B.V. All rights reserved.
298
1
S.S. Kumar, G. Marimuthu / Electronic Notes in Discrete Mathematics 48 (2015) 297–300
Introduction
In this paper we consider only finite and simple undirected bipartite graphs. The vertex and edge sets of a graph G denoted by V (G) and E(G) respectively and we let |V (G)| = p and |E(G)| = q. For graph theoretic notations, we follow [2,3]. Many kinds of labeling have been studied and an excellent survey of graph labelings can be found in [4]. Suppose G is H-decomposable. A total labeling f : V (G) ∪ E(G) → {1, 2, · · · , p + q} is called an H-magic labeling of G if there exists a positive integer k (called magic constant) such that for every copy H in the decom position, v∈V (H) f (v) + e∈E(H) f (e) = k. A graph G that admits such a labeling is called an H-magic decomposable graph. An H-magic labeling f is called a H-E-super magic labeling if f (E(G)) = {1, 2, · · · , q}. A graph that admits a H-E-super magic labeling is called a H-E-super magic decomposable graph. For further details on H-magic type labelings refer [5,6,8,11]. In [9], Subbiah and Pandimadevi introduced the notion of H-E-super magic decomposition of graphs. Yoshimi Ecawa et al. [10] studied the decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components. The notion of star-subgraph was introduced by Akiyama and Kano in [1]. A subgraph F of a graph G is called a star-subgraph if each component of F is a star. Here by a star, we mean a complete bipartite graph of the form K1,m with m ≥ 1. In [7], Marimuthu and Stalin Kumar studied the H-V -super magic decomposition of complete bipartite graphs. A subgraph F of a graph G is called a n-star-subgraph if F ∼ = K1,n with 2 ≤ n < p.
2
Main Result
In this section, we consider the graphs G ∼ = Kn,n and H ∼ = K1,n , where n ≥ 2. 2 Clearly p = 2n and q = n . Theorem 2.1 Suppose {H1 , H2 , · · · , Hn } is an n-star-decomposition of G. Then G is an n-star-E-super magic decomposable graph with magic constant {2n + n(n+1)(3n+1) }. 2 Proof. Let U = {u1 , u2 , · · · , un } and V = {v1 , v2 , · · · , vn } be the bipartition of G. Let {H1 , H2 , · · · , Hn } be an n-star decomposition of G, where each Hi is isomorphic to H with V (Hi ) = {ui , v1 , v2 , · · · , vn } and E(Hi ) = {ui v1 , ui v2 , · · · , ui vn }, for 1 ≤ i ≤ n. Define a total labeling f : V (G) ∪ E(G) → {1, 2, · · · , p + q} by f (ui ) = n2 + (2i − 1) and f (vi ) = n2 + 2i, for all i = 1, 2, · · · , n.
299
S.S. Kumar, G. Marimuthu / Electronic Notes in Discrete Mathematics 48 (2015) 297–300
Case 1. n is even. Now the edge of G are given in Table 1. Table 1 The edge label of an n-star-decomposition of G if n is even. f
v1
v2
v3
...
vn − 1
vn
u1 .. .
n
n+1
3n
...
(n − 1)n
n(n)
...
...
...
...
...
...
uk .. .
n − (k − 1)
n+k
3n − (k − 1)
...
(n − 1)n − (k − 1)
n(n) − (k − 1)
...
...
...
...
...
...
un
1
2n
2n + 1
...
(n − 2)n + 1
(n − 1)n + 1
From Table 1 and from the definition of f , we obtain f (Hk ) = f (uk ) + n n n(n+1)(3n+1) for all k, 1 ≤ k ≤ n. i=1 f (vi ) + i=1 f (uk vi ) = 2n + 2 Case 2. n is odd. Now the edge of G are given in Table 2. From Table 2 and from definition Table 2 The edge label of an n-star-decomposition of G if n is odd. f
v1
v2
...
vn − 1
u1
n
n+1
...
(n − 3)n + +2( n+1 2
n+1 2
vn − 1
vn
(n − 1)n
(n − 1)n + 1
− 1)
.. .
...
...
...
...
...
...
uk
(n + 1) − k
n+k
...
(n − 3)n
(n − 2)n
(n − 1)n
+(n + 1) − k
+(n + 1) − k
+(n + 1) − k
...
...
...
.. . un
... 1
... 2n
... ...
(n − 3)n + 1
(n − 3)n +
n+1 2
+1
n(n)
of f , we obtain , 1 ≤ k ≤ n. f (Hk ) = f (uk ) + ni=1 f (vi ) + ni=1 f (uk vi ) = 2n + n(n+1)(3n+1) 2 Hence the graph G is an n-star-E-super magic decomposable graph. 2
References [1] J. Akiyama and M. Kano, Path factors of a graph, Graphs and Applications, Wiley, Newyork, (1984).
300
S.S. Kumar, G. Marimuthu / Electronic Notes in Discrete Mathematics 48 (2015) 297–300
[2] G. Chartrand, L. Lesniak, Graphs and Digraphs, 3rd Edition, Chapman and Hall, Boca Roton, London, Newyork, Washington, D.C (1996). [3] G. Chartrand and P. Zhang, Chromatic Graph Theory, Chapman and Hall, CRC, Boca Roton, (2009). [4] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin., 16 (2013), #DS6. [5] A. Guti´ errez, A. Llad´ o, Magic coverings, J. Combin. Math. Combin. Comput., 55 (2005), 43–56. [6] A. Llad´ o, J.Moragas, Cycle-magic graphs, Discrete Math., 307 (2007), 2925– 2933. [7] G. Marimuthu, S. Stalin Kumar, H-V -super magic decomposition of complete bipartite graphs, Communicated. [8] A. A. G. Ngurah, A. N. M. Salman, L. Susilowati, H-Supermagic labeling of graphs, Discrete Math., 310 (2010),1293–1300. [9] P. Subbiah, J. Pandimadevi, H-E-super magic decompositions of graphs, Electronic Journal of Graph Theory and and Applications 2(2) (2014), 115– 128. [10] Yoshimi Egawa, Masatsugu Urabe, Toshihito Fukuda and Seiichiro Nagoya, A Decomposition of Complete bipartite graphs into edge-disjoint subgraphs with star components, Discrete Math., 58 (1986), 93–95. [11] Zhihe Liang, Cycle-super magic decompositions of complete multipartite graphs, Discrete Math., 312 (2012), 3342–3348.