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International Journal of Combinatorics
Volume 2013 (2013), Article ID 907249, 4 pages
http://dx.doi.org/10.1155/2013/907249
Research Article

Sunlet Decomposition of Certain Equipartite Graphs

1Department of Mathematics, University of Agriculture, Makurdi 970001, Nigeria
2Department of Mathematics, University of Ibadan, Ibadan 200001, Nigeria

Received 28 September 2012; Accepted 5 February 2013

Academic Editor: Chris A. Rodger

Copyright © 2013 Abolape D. Akwu and Deborah O. A. Ajayi. 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

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 sunlet graph of length , for a prime .

We extend these results by considering the decomposition of into sunlet graphs and prove the following result.

Let , , and be even integers. The graph can be decomposed into sunlet graph of length if and only if divides , the number of edges in .

2. Proof of the Result

To prove the result, we need the following.

Lemma 1 (see [10]). For , decomposes .

Lemma 2. For any integer and a positive even integer , the graph has a decomposition into sunlet graph , for .

Proof   
Case 1 ( is even). First observe that can be decomposed into sunlet graphs with vertices. Now, set and decompose into cycles . To decompose into -cycles , denote vertices in th part of by for , and create base cycles . Next, combine these base cycles into one cycle by replacing each edge with . To create the remaining cycles , we apply mappings for defined on the vertices as follows.
Subcase  1.1 ( odd). Consider This is the desired decomposition into cycles .
Subcase  1.2 ( even). Consider This is the desired decomposition into cycles .
Now take each cycle , and make it back into . Each decomposes into sunlet graphs (by Lemma 1), and we have decomposing into sunlet graphs with length for even. Note that Case 2 ( is odd)
Subcase 2.1 (). Set . First create cycles in as in Case . Then, take complete tripartite graph with partite sets for and and decompose it into triangles using well-known construction via Latin square, that is, construct Latin square and consider each element in the form where denotes the row, denotes the column, and denotes the entry with . Each cycle is of the form ,  ,  . Then, for every triangle , replace the edge in each , by the edges and to obtain cycles . Therefore, . Now take each cycle , make it into , and by Lemma 1, has a decomposition into sunlet graphs .
Subcase 2.2 (). Set . The graph decomposes into Hamilton cycle by [11]. Next, make each cycle into . Each graph decomposes into sunlet graph by Lemma 1.

Theorem 3. Let ,   be positive integers satisfying , then decomposes .

Proof. Let the partite sets (layers) of the -partite graph be . Set . Obtain a new graph from as follows.
Identify the subsets of vertices , for and into new vertices , and identify the subset of vertices for and into new vertices and two of these vertices , where , are adjacent if and only if the corresponding subsets of vertices in induce . The resulting graph is isomorphic to . Next, decompose into cycles as follows: where ,   are calculated modulo .
To construct the remaining cycles, apply mapping defined on the vertices.
Subcase 1.1 ( odd in each cycle). Consider This is the desired decomposition of into cycles .
Subcase 1.2 ( even in each cycle). Consider This is the desired decomposition of into cycles .
By lifting back these cycles of to , we get edge-disjoint subgraphs isomorphic to . Obtain a new graph again from as follows.
For each , identify the subsets of vertices , where into new vertices , and two of these vertices are adjacent if and only if the corresponding subsets of vertices in induce . The resulting graph is isomorphic to . Then, decompose into cycles . Each decomposes into cycles by [12]. By lifting back these cycles of to , we get edge-disjoint subgraph isomorphic to . Finally, each decomposes into two sunlet graphs (by Lemma 1), and we have decomposing into sunlet graphs as required.

Theorem 4 (see [12]). The cycle decomposes for every even .

Theorem 5 (see [12]). If and are odd integers, then decomposes .

Theorem 6. The sunlet graph decomposes if and only if either one of the following conditions is satisfied. (1)is a positive odd integer, and is a positive even integer. (2),   are positive even integers with .

Proof. (1) Set , where is a positive integer. Let the partite sets (layers) of the -partite graph be . For each , where , identify the subsets of vertices , for into new vertices , and two of these vertices are adjacent if and only if the corresponding subsets of vertices in induce . The resulting graph is isomorphic to . Then, decompose into cycles , where is a positive integer.
Now, by Theorems 4 and 5.
By lifting back these -cycles of to , we get edge-disjoint subgraphs isomorphic to . Each copy of decomposes into sunlet graphs of length (by Lemma 1), and we have decomposing into sunlet graphs of length as required.
(2) Set , where is an even integer since .
Obtain a new graph from the graph as in Case . By Theorem 4, . By lifting back these -cycles of to , we get edge-disjoint subgraphs isomorphic to . Each copy of decomposes into sunlet graph of length (by Lemma 1). Therefore, as required.

Remark 7. In [10], it was shown that This, coupled with Lemma 1, gives the following.

Theorem 8 (see [10]). The graph decomposes into sunlet graphs for any positive integer .

Lemma 9 (see [3]). Let be an even integer. Then, is -decomposable.

Lemma 10 (see [3]). Let and be integers with odd, ,  , and . Then, is -decomposable.

Lemma 11 (see [3]). Let and be integers with odd, ,  , and . Then, is -decomposable.

We can now prove the major result.

Theorem 12. For any even integers ,  , and , the sunlet graph decomposes if and only if .

Proof. The necessity of the condition is obvious, and so we need only to prove its sufficiency. We split the problem into the following two cases.
Case 1 ()
Subcase 1.1 (). Cycle decomposes by Lemma 9, and we have Each graph decomposes into sunlet graph , where by Lemma 2, and we have decomposing into sunlet graph , where .
Subcase 1.2 (). First, consider .
Cycle decomposes by Lemma 9, and we have Now, sunlet graph by Theorem 3, and hence sunlet graph decomposes .
Also, consider .
Suppose . Cycle decomposes by Lemma 10, and we have Now, sunlet graph decomposes by Theorem 8, and we have decomposing into sunlet graph of length .
Case 2 ()
Subcase 2.1 (). Suppose , and by Lemma 9, cycle decomposes , and we have Also, sunlet graph decomposes each by Theorem 6, and we have sunlet graph decomposing .
Subcase 2.2 (). Let and an odd integer. Cycle decomposes , by Lemmas 9, 10, and 11, and we have Now, each decomposes into sunlet graph by Theorem 6, and we have decomposing into sunlet graph as required.
Subcase 2.3 (). Set , where is any positive integer, then by Subcases 2.1 and 2.2, we have Each graph decomposes into sunlet graph by Remark 7, and we have decomposing into sunlet graph .

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