Induced Cycle Decomposition of Graphs

Transcription

1 Applied Mathematical Sciences, Vol. 9, 2015, no. 84, HIKARI Ltd, Induced Cycle Decomposition of Graphs Rosalio G. Artes, Jr. Department of Mathematics and Statistics College of Science and Mathematics, MSU - Iligan Institute of Technology Andres Bonifacio Avenue, Tibanga, 9200 Iligan City, Philippines Raymund A. Indangan College of Education J.H. Cerilles State College Mati, San Miguel, Zamboanga del Sur, Philippines Copyright c 2015 Rosalio G. Artes, Jr. and Raymund A. Indangan. This article is 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 In this paper, we determined the induced k-cycle decomposability number of some special graphs. In addtition, wwe characterized graphs with induced k-cycle decomposability number equal to n. Some realization theorems are also established. Specifically, it is shown in this study that for a graph G of order n, its induced k-cycle decomposability number is equal to n if and only if its vertex set is an induced k-cycle decomposable set. For every pair of positive integers n and k where k n and k divides n, there exists a connected graph G of order n such that its induced k-cycle decomposability number is exactly equal to n. It is also shown in this study that the induced k-cycle decomposability number of a graph is at least equal to the induced k-cycle decomposability number of its subgraph. Mathematics Subject Classification: 05C0, 05C8 Keywords: induced k-cycle, induced k-cycle decomposable set, induced k-cycle decomposability number

2 4166 R.G. Artes, Jr. and R.A. Indangan 1 Introduction Decomposition of graphs into induced k-cycles is perhaps first introduced in this paper. In 2007, Artes and Luga [1] introduced the concept of induced m-path decomposability number of graphs. They have generated results on the induced m-path decomposability number of graphs resulting from the join and the corona of two graphs. Moreover, they established that for every pair of positive integers n and k where k is at most n and n divisible by k, there exists a graph G of order n whose induced m-path decomposability number exactly equal to k. Given a cyclic graph G of order n and k be a positive integer. An induced k-cycle of G of order k is a cycle in G of order k induced by a subset of V (G).A subset S of V (G) is an induced k-cycle decomposable set if S can be partitioned into subsets each of which induces a k-cycle in G. That is, there exists a collection C = {S 1, S 2, S,..., S r } of subsets of S such that S i is a k-cycle in G for each i = 1, 2,,..., r, S i S j = for i j, and r S i = S. The induced k-cycle decomposability number of G is given by i=1 Γ k (G) = max{ S : S is an induced k-cycle decomposable subset of V (G)}. For convenience, whenever a graph G has no cycle of order k, we define its k-cycle decomposability number to be equal to 0. 2 Preliminary Results and Some Characterizations This section presents some preliminary results and basic properties which will be useful in establishing the induced k-cycle decomposability number of some graphs. The following remark follows directly from the definition of the induced k-cycle decomposability number of a graph. Theorem 2.1 Let G be a graph of order n. Then 1. For every k, 0 Γ k (G) n. 2. If C = {S 1, S 2, S,..., S r } is an induced k-cycle decomposition of a subset S of V (G), then S = rk.

3 Induced cycle decomposition of graphs 4167 The following theorem characterizes all graphs whose induced k-cycle decomposability number equal to its order. Theorem 2.2 Let G be a graph of order n and k be a positive integer such that k n. Then Γ k (G) = n if and only if V (G) = n is a k-cycle decomposable set. Proof : Assume that Γ k (G) = n. Then n is the cardinality of the maximum k-cycle decomposable subset of V (G). Thus, S = V (G) is an induced k-cycle decomposable set. Conversely, assume that V (G) is an induced k-cycle decomposable set. By definition. Γ k (G) V (G) = n. By Theorem 2.1, Γ k (G) n. Accordingly, Γ k (G) = n. A direct consequence of Theorem 2.2 is obtained when k = n. Corollary 2. Let G be a graph of order n. Then, Γ n (G) = n if and only if G is a cycle. Proof : Let G be a graph of order n. If Γ n (G) = n, then V (G) is an induced n-cycle decomposable set by Theorem 2.2. Moreover, the only decomposition of V (G) into induced n-cycle is the collection C = {V (G)}. Hence, V (G) must be the cycle of order n. Conversely, assume that G is a cycle. Let S = V (G) = V (C n ). Then S is an induced n-cycle decomposable set. Thus, Γ n (G) S = V (G) = n. By Theorem 2.1, Γ n (G) n. Accordingly, Γ n (G) = n. The existence of a connected graph G of order n whose induced k-cycle decomposability number equal to n is formally stated in the following theorem. Theorem 2.4 For every pair of positive integers n and k where k n and k divides n, there exists a connected graph G of order n such that Γ k (G) = n. Proof : Let k be a positive integer such that k n. Since k divides n, there exist r Z such that n = rk. Construct r disjoint cycles each of order k. Let G be the graph obtained by adding edges to connect these cycles in a series. Then V (G) is an induced k-cycle decomposable set. Hence, by Theorem 2.2, Γ n (G) = n

4 4168 R.G. Artes, Jr. and R.A. Indangan The next result establishes the relationship between the induced k-cycle decomposability number of a graph and the induced k-cycle decomposability number of its subgraph. Theorem 2.5 Let G be a graph and H be a subgraph of G. Then for every positive integer k with the property k V (G), Γ k (H) Γ k (G). Proof : Let S G and S H be the induced k-cycle decomposable subsets of V (G) and V (H), respectively, such that Γ k (G) = S G and Γ k (H) = S H. Since H is a subgraph of G, V (H) V (G) and E(H) E(G). It is clear that S H S G. Hence, Γ k (H) = S H S G = Γ k (G). The following section establishes the induced k-cycle decomposability number of cycles and complete graphs. 2.1 Decomposability Number of Cycles and Complete Graphs The induced k-cycle decomposability number of cycles is established in the following theorem. Theorem 2.6 Let n be a positive integer. Then, { n, if k = n Γ k (C n ) = 0, if k < n Proof : By Corollary 2., Γ k (C n ) = n if k = n. It is clear that C n has no cycle of order less than k. By convention, for k < n, Γ k (C n ) = 0. For the complete graph K n, its induced k-cycle decomposability number is given in the following theorem. Theorem 2.7 For every n, { n Γ k (K n ) =, if k = 0, otherwise Proof : Let V (K n ) = {v 1, v 2,..., v n }. Then a subset S of V (K n ) induces a cycle in V (K n ) if and only if S =. Hence, Γ k (K n ) = 0 by convention whenever k. For k =, let S be the union of disjoint -subsets of V (G). Then S is a -cycle decomposable set. Thus, Γ (K n ) n. If divides n, then by Theorem 2.4, Γ (K n ) = n. Otherwise, Γ (K n ) n. Accordingly, Γ (K n ) = n.

5 Induced cycle decomposition of graphs 4169 References [1] Artes, Rosalio Jr. G. and Luga, Mary Joy F. Induced m-path Decomposition of Graphs, unpublished. [2] J. A. Bondy and V. S. R. Murty, Graph Theory with applications, London: Macmillan Press, [] G. Chartrand, E. Curtiss and P. Zhang, The Convexity Sets in Graphs, Congressus Numeratium, 16 (1999). [4] G. Chartrand and L. Lesniak, Graphs and Digraphs, New York: Chapman and Hall, [5] G. Chartrand and O. R. Oellermann, Applied and Algorithmic Graph Theory, New York: McGraw-Hill, Inc., 199. [6] F. Harary, Graph Theory, Reading MA: Addison-Wesley, Received: April 4, 2015; Published: June 4, 2015