Sunlet Decomposition of Certain Equipartite Graphs

Let stand for the sunlet graph which is a graph that consists of a cycle and an edge terminating in a vertex of degree one attached to each vertex of cycle . The necessary condition for the equipartite graph to be decomposed into for is that the order of must divide , the order of . In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques. 1. Introduction Let ,？？ ,？？ denote cycle of length , complete graph on vertices, and complement of complete graph on vertices. For even, denotes the multigraph obtained by adding the edges of a 1-factor to , thus duplicating edges. The total number of edges in is . The lexicographic product, , of graphs and , is the graph obtained by replacing every vertex of by a copy of and every edge of by the complete bipartite graph . For a graph , an -decomposition of a graph , , is a set of subgraphs of , each isomorphic to , whose edge set partitions the edge set of . Note that for any graph and and any positive integer , if then . Let be a graph of order and any graph. The corona (crown) of with , denoted by , is the graph obtained by taking one copy of and copies of and joining the th vertex of with an edge to every vertex in the th copy of . A special corona graph is , that is, a cycle with pendant points which has vertices. This is called sunlet graph and denoted by , . Obvious necessary condition for the existence of a -cycle decomposition of a simple connected graph is that has at least vertices (or trivially, just one vertex), the degree of every vertex in is even, and the total number of edges in is a multiple of the cycle length . These conditions have been shown to be sufficient in the case that is the complete graph , the complete graph minus a -factor [1, 2], and the complete graph plus a -factor [3]. The study of cycle decomposition of was initiated by Hoffman et al. [4]. The necessary and sufficient conditions for the existence of a -decomposition of , where ( is prime) that (i) is even and (ii) divides , were obtained by Manikandan and Paulraja [5, 6]. Similarly, when is a prime, the necessary and sufficient conditions for the existence of a -decomposition of were given by Smith [7]. For a prime number , Smith [8] showed that -decomposition of exists if the obvious necessary conditions are satisfied. In [9], Anitha and Lekshmi proved that the complete graph and the complete bipartite graph for even have decompositions into sunlet graph . Similarly, in [10], it was shown that the complete equipartite graph has a decomposition into