Graceful Labeling for Complete Bipartite Graphs
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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 103, HIKARI Ltd, Graceful Labeling for Complete Bipartite Graphs V. J. Kaneria Department of Mathematics Saurashtra University, Rajkot , India H. M. Makadia Govt. Engineering College, Rajkot , India M. M. Jariya V. V. P. Engineering College, Rajkot , India Meera Meghapara Om Engineering College, Junagadh , India Copyright c 2014 V. J. Kaneria, H. M. Makadia, M. M. Jariya and Meera Meghapara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper contains some results on graceful labeling. We prove that the path union of complete bipartite graph and join sum of complete bipartite graphs are graceful. We also prove that star of complete bipartite graph is graceful. Mathematics Subject Classification: 05C78 Keywords: Complete bipartite graph, path union, join sum of graphs, star of a graph
2 5100 V. J. Kaneria, H. M. Makadia, M. M. Jariya and Meera Meghapara 1 Introduction : Let G = (V, E) be a simple, undirected and finite graph with p vertices and q edges. In this work K m,n denotes a complete bipartite graph. For all other terminology and notations we follows Harary (Harary 1972). We will give brief summary of definitions which are useful for this paper. Definition 1.1: A function f is called graceful labeling of a graph G = (V, E) if f : V {0, 1,..., q} is injective and the induce function f : E {1, 2,..., q} defined as f (e = uv) = f(u) f(v) is bijective, e = uv E. A graph G, which admits graceful labeling is called graceful graph. Definition 1.2: Let G be a graph and G 1, G 2,..., G n, n 2 be n copies of graph G. Then the graph obtained by adding an edge from G i to G i+1 (i = 1, 2,..., n 1) is called path union of G. Definition 1.3: Consider t copies of a graph G 0. Then graph G =< G (1) 0 ; G (2) 0 ;... ; G (t) 0 > obtained by joining two copies of the graph G (i) 0 and G (i+1) 0 by a vertex 1 i t 1 is called join sum of graphs. Definition 1.4: A graph obtained by replacing each vertex of star K 1,n by a graph G of n vertices is called star of G and it is denoted by G. The graph G which replaced at the center of K 1,n we call the central copy of G. For detail survey of graph labeling one can refer Gallian (Gallian 2013). Labeled graphs have many diversified applications. The graceful labeling was introduced by Rosa (Rosa 1967, p ). Golomb (Golomb 1972, p ) proved that the complete bipartite graph K m,n is graceful. Barrientos (Barrientos 2005, p ) proved that union of complete bipartite graphs is also graceful. Vaidya et al. (vaidya, Ghodasara, Sweta and Kaneria 2008, p ) introduced a star of a cycle and proved that it is cordial as well as 3 equitable graph. Kaneria and Makadia (Kaneria and Makadia 2012, p )proved that a star of a cycle C n (n 0 (mod 4)) is graceful. In present paper we introduced gracefulness of path union of complete bipartite graph, join sum of complete bipartite graphs and star of a complete bipartite graph. 2 Main Results : Theorem 2.1 : G =< K m1,n 1 ;... ; K mt,nt > the join sum of complete bipartite graphs is graceful, where m 1, n 1,..., m t, n t N. Proof : Let G be the join sum of complete bipartite graphs K m1,n 1,..., K mt,nt, where m 1, n 1,..., m t, n t N. Let u i,j (1 j m i ), v i,j (1 j n i ) be vertices of the complete bipartite graphs K mi,n i, i = 1, 2,..., t. Let w 1, w 2,..., w t 1 be vertices for join sum of the complete bipartite graphs and join vertices (u i,ni, w i ), (w i, u i+1,1 ), i = 1, 2,..., t 1 by an edge to produce join sum of complete bipartite graphs < K m1,n 1 ;... ; K mt,nt >.
3 Graceful labeling for complete bipartite graphs 5101 We define labeling function f : V {0, 1,..., q}, where q = Σ t i=1 m i n i + 2(t 1) as follows: f(u 1,i ) = i 1, i = 1, 2,..., m 1 f(v 1,i ) = q m 1 (i 1), i = 1, 2,..., n 1 and f(w j 1 ) = q Σ j 1 l=1 m l(n l 1) 2j + 3, f(u j,i ) = Σ j 1 l=1 m l + (i 1), i = 1, 2,..., m j f(v j,i ) = q Σ j 1 l=1 m l(n l 1) 2(j 1) m j (i 1), i = 1, 2,..., n j ; j = 2, 3,..., t. Above labeling pattern give rise graceful labeling to < K m1,n 1 ;... ; K mt,nt >. Illustration 2.2: G =< K 4,3 ; K 3,3 ; K 2,4 > and its graceful labeling shown in figure 1. Figure 1 Theorem 2.3 : Path union of complete bipartite graphs K m1,n 1,..., K mt,n t is graceful, where m 1, n 1,..., m t, n t N. Proof : Let G be a path union of complete bipartite graphs K m1,n 1,..., K mt,nt, where m 1, n 1,..., m t, n t N. To produce the path union of this graphs we join u i,mi with v i+1,1 by an edge, i = 1, 2,..., t 1, where u i,j (1 j m i ), v i,j (1 j n i ) are vertices of the complete bipartite graph K mi,n i, i = 1, 2,..., t. We define labeling function f : V m i n i + (t 1) as follows: {0, 1,..., q}, where q = Σ t i=1 f(u 1,i ) = i 1, i = 1, 2,..., m 1 f(v 1,i ) = q m 1 (i 1), i = 1, 2,..., n 1 f(u j,i ) = Σ j 1 l=1 m l + (i 1), i = 1, 2,..., m j f(v j,i ) = q Σ j 1 l=1 m l(n l 1) (j 1) m j (i 1), i = 1, 2,..., n j ; j = 2, 3,..., t. Above labeling pattern give rise graceful labeling to required graph.
4 5102 V. J. Kaneria, H. M. Makadia, M. M. Jariya and Meera Meghapara Illustration 2.4: Path union of K 3,2 ; K 4,3 ; K 1,4 and its graceful labeling shown in figure 2, in which edge labels are descending order at lower set of vertices. Figure 2 Theorem 2.5 : K m,n, the star of a complete bipartite graph is graceful. Proof : Let u 0,i (1 i m), v 0,j (1 j n) be vertices of central copy of Km,n and u l,i (1 i m), v l,j (1 j n) be vertices of other copies of K m,n, l = 1, 2,..., m + n. We define labeling function f : V {0, 1,..., q}, where q = (m + n + 1)mn + m + n as follows: f(u 0,i ) = i 1, i = 1, 2,..., m f(v 0,j ) = q (i 1)m, j = 1, 2,..., n f(u 1,i ) = q (mn + 1) + i, i = 1, 2,..., m f(v 1,i ) = jm, j = 1, 2,..., n f(u l,i ) = f(u l 2,i ) + (mn + 1), i = 1, 2,..., m f(v l,j ) = f(v l 2,j ) (mn + 1), j = 1, 2,..., n; l = 2, 3,..., m + n. Above defined labeling function f give rise edge labels 1, 2,..., mn, mn+ 2, mn + 3,..., 2mn + 1, 2mn + 3, 2mn + 4,..., 3mn + 2,..., (mn + 1)(m + n) + 1,..., q to all the copies of K m,n. To make Km,n as graceful graph mn + 1, 2(mn + 1),..., (m + n)(mn + 1) edge labels to be require. Now we see that the difference of vertex labels for the central copy K (0) m,n with its other copies K (i) m,n, 1 i m + n is precisely following sequence. (m + n)(mn + 1) mn + 1 (m + n 1)(mn + 1) 2(mn + 1)..
5 Graceful labeling for complete bipartite graphs 5103 ( m+n )(mn + 1), when m + n 0 (mod 2) 2 or )(mn + 1), when m + n 1 (mod 2) 2 ( m+n+1 )(mn + 1). 2 ( m+n 1 Using above sequence we can produce required edge labels by joining corresponding vertex of K m,n (0) with its other copies of K m,n, (i) i = 1, 2,..., m + n. Thus Km,n admits a graceful labeling and so it is a graceful graph. Illustration 2.6: K3,2 and its graceful labeling shown in figure 3. Figure 3 3 Concluding Remarks : We discussed here graceful labeling of some graphs obtained by complete bipartite graphs. Present work contributes three new results. Labeling pattern is demonstrated by means of illustrations, which provide better understanding of derived results. Acknowledgement: Authors of this paper would like to thanks reviewers for their valuable suggestions.
6 5104 V. J. Kaneria, H. M. Makadia, M. M. Jariya and Meera Meghapara References [1] C. Barrientos, The gracefulness of unions of cycles and complete bipartite graphs, J. Combin. Math. Combin. Comput. 52 (2005), [2] J. A. Gallian, The Electronics Journal of Combinatorics, 19, DS6(2013). [3] S. W. Golomb, How to number a graph, in Graph Theory and Computing, R. C. Read, ed., Academic Press, New York, (1972) [4] F. Harary, Graph theory Addition Wesley, Massachusetts, [5] V. J. Kaneria and H. M. Makadia, Some Graceful Graphs J. of Math. Research, 4 (1), (2012) [6] A. Rosa, On certain valuation of graph, Theory of Graphs (Rome, July 1966), Goden and Breach, N. Y. and Paris, 1967, [7] S. K. Vaidya, G. V. Ghodasara, Sweta Srivastav and V. J. Kaneria Cordial and 3 equitable labeling of Star of a Cycle, Mathematics Today, 24 (2008), Received: June 7, 2014
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