Hindawi Publishing Corporation International Journal of Combinatorics Volume 2013, Article ID 907249, 4 pages http://dx.doi.org/10.1155/2013/907249 Research Article Sunlet Decomposition of Certain Equipartite Graphs Abolape D. Akwu1 and Deborah O. A. Ajayi2 1 2 Department of Mathematics, University of Agriculture, Makurdi 970001, Nigeria Department of Mathematics, University of Ibadan, Ibadan 200001, Nigeria Correspondence should be addressed to Deborah O. A. Ajayi; adelaideajayi@yahoo.com Received 28 September 2012; Accepted 5 February 2013 Academic Editor: Chris A. Rodger Copyright © 2013 A. D. Akwu and D. 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. Let πΏ 2π 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 πΆπ . The necessary condition for the equipartite graph πΎπ + πΌ ∗ πΎπ to be decomposed into πΏ 2π for π ≥ 2 is that the order of πΏ 2π must divide π2 π2 /2, the order of πΎπ + πΌ ∗ πΎπ . In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques. 1. Introduction Let πΆπ , πΎπ , πΎπ denote cycle of length π, complete graph on π vertices, and complement of complete graph on π vertices. For π even, πΎπ + πΌ denotes the multigraph obtained by adding the edges of a 1-factor to πΎπ , thus duplicating π/2 edges. The total number of edges in πΎπ + πΌ is π2 /2. The lexicographic product, πΊ ∗ π», of graphs πΊ and π», is the graph obtained by replacing every vertex of πΊ by a copy of π» and every edge of πΊ by the complete bipartite graph πΎ|π»|,|π»| . For a graph π», an π»-decomposition of a graph πΊ, π» | πΊ, is a set of subgraphs of πΊ, each isomorphic to π», whose edge set partitions the edge set of πΊ. Note that for any graph πΊ and π» and any positive integer π, if π» | πΊ then (π»∗πΎπ ) | (πΊ∗πΎπ ). Let πΊ be a graph of order π and π» any graph. The corona (crown) of πΊ with π», denoted by πΊ β π», is the graph obtained by taking one copy of πΊ and π copies of π» and joining the πth vertex of πΊ with an edge to every vertex in the πth copy of π». A special corona graph is πΆπ βπΎ1 , that is, a cycle with pendant points which has 2π vertices. This is called sunlet graph and denoted by πΏ π , π = 2π. Obvious necessary condition for the existence of a π-cycle decomposition of a simple connected graph πΊ is that πΊ has at least π vertices (or trivially, just one vertex), the degree of every vertex in πΊ is even, and the total number of edges in πΊ is a multiple of the cycle length π. These conditions have been shown to be sufficient in the case that πΊ is the complete graph πΎπ , the complete graph minus a 1-factor πΎπ − πΌ [1, 2], and the complete graph plus a 1-factor πΎπ + πΌ [3]. The study of cycle decomposition of πΎπ ∗πΎπ was initiated by Hoffman et al. [4]. The necessary and sufficient conditions for the existence of a πΆπ -decomposition of πΎπ ∗ πΎπ , where π ≥ 5 (π is prime) that (i) π(π − 1) is even and (ii) π divides π(π−1)π2 , were obtained by Manikandan and Paulraja [5, 6]. Similarly, when π ≥ 3 is a prime, the necessary and sufficient conditions for the existence of a πΆ2π -decomposition of πΎπ ∗ πΎπ were given by Smith [7]. For a prime number π ≥ 3, Smith [8] showed that πΆ3π -decomposition of πΎπ ∗πΎπ exists if the obvious necessary conditions are satisfied. In [9], Anitha and Lekshmi proved that the complete graph πΎπ and the complete bipartite graph πΎπ,π for π even have decompositions into sunlet graph πΏ π . Similarly, in [10], it was shown that the complete equipartite graph πΎπ ∗πΎπ has a decomposition into sunlet graph of length 2π, for a prime π. We extend these results by considering the decomposition of πΎπ + πΌ ∗ πΎπ into sunlet graphs and prove the following result. Let π ≥ 2, π > 2, and π ≥ 6 be even integers. The graph πΎπ + πΌ ∗ πΎπ can be decomposed into sunlet graph of length π if and only if π divides π2 π2 /2, the number of edges in πΎπ + πΌ ∗ πΎπ . 2 International Journal of Combinatorics Theorem 3. Let π, π be positive integers satisfying π, π ≡ 0 (mod 4), then πΏ π decomposes πΆπ ∗ πΎπ . 2. Proof of the Result To prove the result, we need the following. Lemma 1 (see [10]). For π ≥ 3, πΏ 2π decomposes πΆπ ∗ πΎ2 . Lemma 2. For any integer π > 2 and a positive even integer π, the graph πΆπ ∗ πΎπ has a decomposition into sunlet graph πΏ π , for π = ππ. Proof Case 1 (π is even). First observe that πΆπ ∗ πΎ2 can be decomposed into 2 sunlet graphs with 2π vertices. Now, set π = 2π‘ and decompose πΆπ ∗ πΎπ‘ into cycles πΆππ‘ . To decompose πΆπ ∗ πΎπ‘ into π‘-cycles πΆππ‘ , denote vertices in πth part of πΆπ ∗ πΎπ‘ by π₯π,π for π = 1, . . . , π‘, π = 1, 2, . . . , π and create π‘ base cycles π₯1,π π₯2,π π₯3,π ⋅ ⋅ ⋅ π₯π−1,π π₯π,π . Next, combine these base cycles into one cycle πΆππ‘ by replacing each edge π₯1,π π₯2,π with π₯1,π π₯2,π+1 . To create the remaining cycles πΆππ‘ , we apply mappings ππ for π = 0, 1, . . . , π‘ − 1 defined on the vertices as follows. Subcase 1.1 (π odd). Consider ππ (π₯π,π ) = π₯π,π . (1) π π 3π π + 1, + 1, + 1, . . . , π − + 1, 4 2 4 4 π= π + π, 4 (4) where π, π are calculated modulo π. To construct the remaining cycles, apply mapping π defined on the vertices. π (π₯π,π ) = π₯π,π+1 . Subcase 1.2 (π even). Consider (2) This is the desired decomposition into cycles πΆππ‘ . Now take each cycle πΆππ‘ , and make it back into πΆππ‘ ∗ πΎ2 . Each πΆππ‘ ∗ πΎ2 decomposes into 2 sunlet graphs πΏ 2ππ‘ (by Lemma 1), and we have πΆπ ∗ πΎπ decomposing into sunlet graphs with length ππ for π even. Note that πΆπ ∗ πΎ2π‘ = (πΆπ ∗ πΎπ‘ ) ∗ πΎ2 . π = 1, Subcase 1.1 (π odd in each cycle). Consider This is the desired decomposition into cycles πΆππ‘ . ππ (π₯π,π ) = π₯π,π+π . Proof. Let the partite sets (layers) of the π-partite graph πΆπ ∗ πΎπ be π1 , π2 , . . . , ππ . Set π = 2π‘. Obtain a new graph from πΆπ ∗ πΎπ as follows. Identify the subsets of vertices {π₯π,π }, for 1 ≤ π ≤ π and 1 ≤ π ≤ π/2 into new vertices π₯π1 , and identify the subset of vertices {π₯π,π } for 1 ≤ π ≤ π and π/2 + 1 ≤ π ≤ π into new vertices π₯π2 and two of these vertices π₯ππ , where π = 1, 2, are adjacent if and only if the corresponding subsets of vertices in πΆπ ∗ πΎπ induce πΎπ‘,π‘ . The resulting graph is isomorphic to πΆπ ∗ πΎ2 . Next, decompose πΆπ ∗ πΎ2 into cycles πΆπ/2 as follows: π₯π,1 π₯π+1,1 , . . . , π₯π,1 π₯π−1,2 , . . . , π₯π+1,2 , π₯π,1 (3) Case 2 (π is odd) Subcase 2.1 (π ≡ 2 (mod 4)). Set π = 2π‘. First create π‘ cycles πΆ(π−1)π‘ in πΆπ−1 ∗ πΎπ‘ as in Case 1. Then, take complete tripartite graph πΎπ‘,π‘,π‘ with partite sets ππ = {π₯π,π } for π = 1, π−1, π and π = 1, . . . , π‘ and decompose it into triangles using well-known construction via Latin square, that is, construct π‘ × π‘ Latin square and consider each element in the form (π, π, π) where π denotes the row, π denotes the column, and π denotes the entry with 1 ≤ π, π, π ≤ π‘. Each cycle is of the form π₯(1,π) , π₯(π−1,π) , π₯(π,π) . Then, for every triangle π₯1,π π₯π−1,π π₯π,π , replace the edge π₯1,π π₯π−1,π in each πΆ(π−1)π‘ , by the edges π₯π−1,π π₯π,π and π₯π,π π₯1,π to obtain cycles πΆππ‘ . Therefore, πΆππ‘ | πΆπ ∗ πΎπ‘ . Now take each cycle πΆππ‘ , make it into πΆππ‘ ∗ πΎ2 , and by Lemma 1, πΆππ‘ ∗ πΎ2 has a decomposition into sunlet graphs πΏ 2ππ‘ = πΏ π . Subcase 2.2 (π ≡ 0 (mod 4)). Set π = 2π‘. The graph πΆπ ∗ πΎπ‘ decomposes into Hamilton cycle πΆππ‘ by [11]. Next, make each cycle πΆππ‘ into πΆππ‘ ∗ πΎ2 . Each graph πΆππ‘ ∗ πΎ2 decomposes into sunlet graph πΏ 2ππ‘ by Lemma 1. (5) This is the desired decomposition of πΆπ ∗ πΎ2 into cycles πΆπ/2 . Subcase 1.2 (π even in each cycle). Consider π (π₯π,π ) = π₯π,π . (6) This is the desired decomposition of πΆπ ∗ πΎ2 into cycles πΆπ/2 . By lifting back these cycles πΆπ/2 of πΆπ ∗ πΎ2 to πΆπ ∗ πΎ2π‘ , we get edge-disjoint subgraphs isomorphic to πΆπ/2 ∗ πΎπ‘ . Obtain a new graph again from πΆπ/2 ∗ πΎπ‘ as follows. For each π, 1 ≤ π ≤ π‘/2, identify the subsets of vertices π {π₯π,2π−1 , π₯π,2π }, where 1 ≤ π ≤ π/2 into new vertices π₯π , π and two of these vertices π₯π are adjacent if and only if the corresponding subsets of vertices in πΆπ/2 ∗πΎπ‘ induce πΎ2,2 . The resulting graph is isomorphic to πΆπ/2 ∗πΎπ‘/2 . Then, decompose πΆπ/2 ∗ πΎπ‘/2 into cycles πΆπ/2 . Each πΆπ/2 ∗ πΎπ‘/2 decomposes into cycles πΆπ/2 by [12]. By lifting back these cycles πΆπ/2 of πΆπ/2 ∗ πΎπ‘/2 to πΆπ/2 ∗πΎπ‘ , we get edge-disjoint subgraph isomorphic to πΆπ/2 ∗ πΎ2 . Finally, each πΆπ/2 ∗ πΎ2 decomposes into two sunlet graphs πΏ π (by Lemma 1), and we have πΆπ ∗ πΎπ decomposing into sunlet graphs πΏ π as required. Theorem 4 (see [12]). The cycle πΆπ decomposes πΆπ ∗ πΎπ for every even π > 3. Theorem 5 (see [12]). If π and π ≥ 3 are odd integers, then πΆπ decomposes πΆπ ∗ πΎπ . International Journal of Combinatorics 3 Theorem 6. The sunlet graph πΏ π decomposes πΆπ ∗ πΎπ if and only if either one of the following conditions is satisfied. (1) π is a positive odd integer, and π is a positive even integer. (2) π, π are positive even integers with π ≡ 0 (mod 4). Proof. (1) Set π = 2π‘, where π‘ is a positive integer. Let the partite sets (layers) of the π-partite graph πΆπ ∗ πΎπ be π1 , π2 , . . . , ππ . For each π, where 1 ≤ π ≤ π‘, identify the subsets of vertices {π₯π,2π−1 , π₯π,2π }, for 1 ≤ π ≤ π into new π π vertices π₯π , and two of these vertices π₯π are adjacent if and only if the corresponding subsets of vertices in πΆπ ∗πΎπ induce πΎ2,2 . The resulting graph is isomorphic to πΆπ ∗ πΎπ‘ . Then, decompose πΆπ ∗πΎπ‘ into cycles πΆπ‘ , where π‘ is a positive integer. Now, πΆπ‘ | πΆπ ∗ πΎπ‘ by Theorems 4 and 5. By lifting back these π‘-cycles of πΆπ ∗ πΎπ‘ to πΆπ ∗ πΎ2π‘ , we get edge-disjoint subgraphs isomorphic to πΆπ‘ ∗ πΎ2 . Each copy of πΆπ‘ ∗ πΎ2 decomposes into sunlet graphs of length 2π‘ (by Lemma 1), and we have πΆπ ∗ πΎπ decomposing into sunlet graphs of length π as required. (2) Set π = 2π‘, where π‘ is an even integer since π ≡ 0 (mod 4). Obtain a new graph πΆπ ∗ πΎπ‘ from the graph πΆπ ∗ πΎπ as in Case 1. By Theorem 4, πΆπ‘ | πΆπ ∗ πΎπ‘ . By lifting back these π‘-cycles of πΆπ ∗ πΎπ‘ to πΆπ ∗ πΎ2π‘ , we get edge-disjoint subgraphs isomorphic to πΆπ‘ ∗ πΎ2 . Each copy of πΆπ‘ ∗ πΎ2 decomposes into sunlet graph of length 2π‘ (by Lemma 1). Therefore, πΏ π | πΆπ ∗ πΎπ as required. Case 1 (π | π) Subcase 1.1 (π > π). Cycle πΆπ decomposes πΎπ + πΌ by Lemma 9, and we have πΆπ ∗ πΎπ | πΎπ + πΌ ∗ πΎπ . (8) Each graph πΆπ ∗ πΎπ decomposes into sunlet graph πΏ π , where π = ππ by Lemma 2, and we have πΎπ + πΌ ∗ πΎπ decomposing into sunlet graph πΏ π , where π > π. Subcase 1.2 (π = π). First, consider π ≡ 0 (mod 4). Cycle πΆπ decomposes πΎπ + πΌ by Lemma 9, and we have πΆπ ∗ πΎπ | πΎπ + πΌ ∗ πΎπ . (9) Now, sunlet graph πΏ π | (πΆπ ∗ πΎπ ) by Theorem 3, and hence sunlet graph πΏ π decomposes πΎπ + πΌ ∗ πΎπ . Also, consider π ≡ 2 (mod 4). Suppose π = 2π‘. Cycle πΆπ/2 decomposes πΎπ + πΌ by Lemma 10, and we have πΆπ/2 ∗ πΎ2π‘ | πΎπ + πΌ ∗ πΎ2π‘ . (10) Now, sunlet graph πΏ π decomposes πΆπ/2 ∗ πΎ2π‘ by Theorem 8, and we have πΎπ + πΌ ∗ πΎπ decomposing into sunlet graph of length π. Case 2 (π | π) Subcase 2.1 (π ≡ 0 (mod 4)). Suppose π = π, and by Lemma 9, cycle πΆπ decomposes πΎπ + πΌ, and we have Remark 7. In [10], it was shown that πΏ 2π ∗ πΎπ can be decomposed into π2 copies of πΏ 2π . 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. (7) This, coupled with Lemma 1, gives the following. Theorem 8 (see [10]). The graph πΆπ ∗ πΎ2π decomposes into sunlet graphs πΏ 2π for any positive integer π. Lemma 9 (see [3]). Let π ≥ 4 be an even integer. Then, πΎπ + πΌ is πΆπ -decomposable. Lemma 10 (see [3]). Let π and π be integers with π odd, π ≡ 2 (mod 4), 3 ≤ π ≤ π < 2π, and π2 ≡ 0 (mod 2π). Then, πΎπ + πΌ is πΆπ -decomposable. Lemma 11 (see [3]). Let π and π be integers with π odd, π ≡ 0 (mod 4), 3 ≤ π ≤ π < 2π, and π2 ≡ 0 (mod 2π). Then, πΎπ + πΌ is πΆπ -decomposable. We can now prove the major result. Theorem 12. For any even integers π ≥ 2, π > 2, and π ≥ 6, the sunlet graph πΏ π decomposes πΎπ + πΌ ∗ πΎπ if and only if π2 π2 /2 ≡ 0 (mod π). πΆπ ∗ πΎπ | πΎπ + πΌ ∗ πΎπ . (11) Also, sunlet graph πΏ π decomposes each πΆπ ∗πΎπ by Theorem 6, and we have sunlet graph πΏ π decomposing πΎπ + πΌ ∗ πΎπ . Subcase 2.2 (π ≡ 2 (mod 4)). Let π = π and π ≤ π an odd integer. Cycle πΆπ decomposes πΎπ + πΌ, by Lemmas 9, 10, and 11, and we have πΆπ ∗ πΎπ | πΎπ + πΌ ∗ πΎπ . (12) Now, each πΆπ ∗ πΎπ decomposes into sunlet graph πΏ π by Theorem 6, and we have πΎπ +πΌ∗πΎπ decomposing into sunlet graph πΏ π as required. Subcase 2.3 (π > π). Set π = π€π, where π€ is any positive integer, then by Subcases 2.1 and 2.2, we have πΏ π ∗ πΎπ€ | (πΎπ + πΌ ∗ πΎπ ) ∗ πΎπ€ . (13) Each graph πΏ π ∗ πΎπ€ decomposes into sunlet graph πΏ π by Remark 7, and we have πΎπ + πΌ ∗ πΎπ decomposing into sunlet graph πΏ π . 4 References [1] B. Alspach and H. Gavlas, “Cycle Decompositions of Kπ and Kπ -I,” Journal of Combinatorial Theory B, vol. 81, no. 1, pp. 77–99, 2001. [2] M. SΜajna, “Cycle decompositions. III. Complete graphs and fixed length cycles,” Journal of Combinatorial Designs, vol. 10, no. 1, pp. 27–78, 2002. [3] M. SΜajna, “Decomposition of the complete graph plus a 1-factor into cycles of equal length,” Journal of Combinatorial Designs, vol. 11, no. 3, pp. 170–207, 2003. [4] D. G. Hoffman, C. C. Lindner, and C. A. Rodger, “On the construction of odd cycle systems,” Journal of Graph Theory, vol. 13, no. 4, pp. 417–426, 1989. [5] R. S. Manikandan and P. Paulraja, “Cπ -decompositions of some regular graphs,” Discrete Mathematics, vol. 306, no. 4, pp. 429– 451, 2006. [6] R. S. Manikandan and P. Paulraja, “C5 -decompositions of the tensor product of complete graphs,” The Australasian Journal of Combinatorics, vol. 37, pp. 285–293, 2007. [7] B. R. Smith, “Decomposing complete equipartite graphs into cycles of length 2p,” Journal of Combinatorial Designs, vol. 16, no. 3, pp. 244–252, 2008. [8] B. R. Smith, “Complete equipartite 3p-cycle systems,” The Australasian Journal of Combinatorics, vol. 45, pp. 125–138, 2009. [9] R. Anitha and R. S. Lekshmi, “N-sun decomposition of complete, complete bipartite and some Harary graphs,” International Journal of Mathematics Sciences, vol. 2, no. 1, pp. 33–38, 2008. [10] A. D. Akwu and D. O. A. Ajayi, “Decomposing equipartite graphs into certain corona graphs of length 2p”. [11] R. Laskar, “Decomposition of some composite graphs into Hamilton Cycles,” in Proceedings of the 5th Hungarian Colloquium keszthely, pp. 705–716, North Holland, 1978. [12] D. FroncΜek, P. KovaΜrΜ, and M. Kubesa, “Decompositions of complete graphs into blown-up cycles C π [2],” Discrete Mathematics, vol. 310, no. 5, pp. 1003–1015, 2010. International Journal of Combinatorics Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Algebra Hindawi Publishing Corporation http://www.hindawi.com Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Journal of Hindawi Publishing Corporation http://www.hindawi.com Stochastic Analysis Abstract and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com International Journal of Mathematics Volume 2014 Volume 2014 Discrete Dynamics in Nature and Society Volume 2014 Volume 2014 Journal of Journal of Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Applied Mathematics Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014