Sunlet Decomposition of Certain Equipartite Graphs

Abstract

Let 𝐿_2𝑛 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 𝐶_𝑛. The necessary condition for the equipartite graph 𝐾_𝑛+𝐼∗𝐾_𝑚 to be decomposed into 𝐿_2𝑛 for 𝑛≥2 is that the order of 𝐿_2𝑛 must divide 𝑛²𝑚²/2, the order of 𝐾_𝑛+𝐼∗𝐾_𝑚. In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques.

1. Introduction

Let 𝐶_𝑟,  𝐾_𝑛,  𝐾_𝑚 denote cycle of length 𝑟, complete graph on 𝑛 vertices, and complement of complete graph on 𝑚 vertices. For 𝑛 even, 𝐾_𝑛+𝐼 denotes the multigraph obtained by adding the edges of a 1-factor to 𝐾_𝑛, thus duplicating 𝑛/2 edges. The total number of edges in 𝐾_𝑛+𝐼 is 𝑛²/2. The lexicographic product, 𝐺∗𝐻, of graphs 𝐺 and 𝐻, is the graph obtained by replacing every vertex of 𝐺 by a copy of 𝐻 and every edge of 𝐺 by the complete bipartite graph 𝐾_{|𝐻|,|𝐻|}.

For a graph 𝐻, an 𝐻-decomposition of a graph 𝐺, 𝐻∣𝐺, is a set of subgraphs of 𝐺, each isomorphic to 𝐻, whose edge set partitions the edge set of 𝐺. Note that for any graph 𝐺 and 𝐻 and any positive integer 𝑚, if 𝐻∣𝐺 then (𝐻∗𝐾_𝑚)∣(𝐺∗𝐾_𝑚).

Let 𝐺 be a graph of order 𝑛 and 𝐻 any graph. The corona (crown) of 𝐺 with 𝐻, denoted by 𝐺⊙𝐻, is the graph obtained by taking one copy of 𝐺 and 𝑛 copies of 𝐻 and joining the 𝑖th vertex of 𝐺 with an edge to every vertex in the 𝑖th copy of 𝐻. A special corona graph is 𝐶_𝑛⊙𝐾₁, that is, a cycle with pendant points which has 2𝑛 vertices. This is called sunlet graph and denoted by 𝐿_𝑞, 𝑞=2𝑛.

Obvious necessary condition for the existence of a 𝑘-cycle decomposition of a simple connected graph 𝐺 is that 𝐺 has at least 𝑘 vertices (or trivially, just one vertex), the degree of every vertex in 𝐺 is even, and the total number of edges in 𝐺 is a multiple of the cycle length 𝑘. These conditions have been shown to be sufficient in the case that 𝐺 is the complete graph 𝐾_𝑛, the complete graph minus a 1-factor 𝐾_𝑛−𝐼 [1, 2], and the complete graph plus a 1-factor 𝐾_𝑛+𝐼 [3].

The study of cycle decomposition of 𝐾_𝑛∗𝐾_𝑚 was initiated by Hoffman et al. [4]. The necessary and sufficient conditions for the existence of a 𝐶_𝑝-decomposition of 𝐾_𝑛∗𝐾_𝑚, where 𝑝≥5 (𝑝 is prime) that (i) 𝑚(𝑛−1) is even and (ii) 𝑝 divides 𝑛(𝑛−1)𝑚², were obtained by Manikandan and Paulraja [5, 6]. Similarly, when 𝑝≥3 is a prime, the necessary and sufficient conditions for the existence of a 𝐶_2𝑝-decomposition of 𝐾_𝑛∗𝐾_𝑚 were given by Smith [7]. For a prime number 𝑝≥3, Smith [8] showed that 𝐶_3𝑝-decomposition of 𝐾_𝑛∗𝐾_𝑚 exists if the obvious necessary conditions are satisfied. In [9], Anitha and Lekshmi proved that the complete graph 𝐾_𝑛 and the complete bipartite graph 𝐾_𝑛,𝑛 for 𝑛 even have decompositions into sunlet graph 𝐿_𝑛. Similarly, in [10], it was shown that the complete equipartite graph 𝐾_𝑛∗𝐾_𝑚 has a decomposition into sunlet graph of length 2𝑝, for a prime 𝑝.

We extend these results by considering the decomposition of 𝐾_𝑛+𝐼∗𝐾_𝑚 into sunlet graphs and prove the following result.

Let 𝑚≥2, 𝑛>2, and 𝑞≥6 be even integers. The graph 𝐾_𝑛+𝐼∗𝐾_𝑚 can be decomposed into sunlet graph of length 𝑞 if and only if 𝑞 divides 𝑛²𝑚²/2, the number of edges in 𝐾_𝑛+𝐼∗𝐾_𝑚.

2. Proof of the Result

To prove the result, we need the following.

Lemma 1 (see [10]). For 𝑟≥3, 𝐿_2𝑟 decomposes 𝐶_𝑟∗𝐾₂.

Lemma 2. For any integer 𝑟>2 and a positive even integer 𝑚, the graph 𝐶_𝑟∗𝐾_𝑚 has a decomposition into sunlet graph 𝐿_𝑞, for 𝑞=𝑟𝑚.

Proof   
Case 1 (𝑟 is even). First observe that 𝐶_𝑟∗𝐾₂ can be decomposed into 2 sunlet graphs with 2𝑟 vertices. Now, set 𝑚=2𝑡 and decompose 𝐶_𝑟∗𝐾_𝑡 into cycles 𝐶_𝑟𝑡. To decompose 𝐶_𝑟∗𝐾_𝑡 into 𝑡-cycles 𝐶_𝑟𝑡, denote vertices in 𝑖th part of 𝐶_𝑟∗𝐾_𝑡 by 𝑥_𝑖,𝑗 for 𝑗=1,…,𝑡, 𝑖=1,2,…,𝑟 and create 𝑡 base cycles 𝑥_1,𝑗𝑥_2,𝑗𝑥_3,𝑗⋯𝑥_{𝑟−1,𝑗}𝑥_𝑟,𝑗. Next, combine these base cycles into one cycle 𝐶_𝑟𝑡 by replacing each edge 𝑥_1,𝑗𝑥_2,𝑗 with 𝑥_1,𝑗𝑥_2,𝑗+1. To create the remaining cycles 𝐶_𝑟𝑡, we apply mappings 𝜙_𝑠 for 𝑠=0,1,…,𝑡−1 defined on the vertices as follows.
Subcase  1.1 (𝑖 odd). Consider 𝜙_𝑠𝑥_𝑖,𝑗=𝑥_𝑖,𝑗. (1) This is the desired decomposition into cycles 𝐶_𝑟𝑡.
Subcase  1.2 (𝑖 even). Consider 𝜙_𝑠𝑥_𝑖,𝑗=𝑥_{𝑖,𝑗+𝑠}. (2) This is the desired decomposition into cycles 𝐶_𝑟𝑡.
Now take each cycle 𝐶_𝑟𝑡, and make it back into 𝐶_𝑟𝑡∗𝐾₂. Each 𝐶_𝑟𝑡∗𝐾₂ decomposes into 2 sunlet graphs 𝐿_2𝑟𝑡 (by Lemma 1), and we have 𝐶_𝑟∗𝐾_𝑚 decomposing into sunlet graphs with length 𝑟𝑚 for 𝑟 even. Note that 𝐶_𝑟∗𝐾_2𝑡=𝐶_𝑟∗𝐾_𝑡∗𝐾₂. (3)Case 2 (𝑟 is odd)
Subcase 2.1 (𝑚≡2(mod4)). Set 𝑚=2𝑡. First create 𝑡 cycles 𝐶_{(𝑟−1)𝑡} in 𝐶_𝑟−1∗𝐾_𝑡 as in Case 1. Then, take complete tripartite graph 𝐾_{𝑡,𝑡,𝑡} with partite sets 𝑋_𝑖={𝑥_𝑖,𝑗} for 𝑖=1,𝑟−1,𝑟 and 𝑗=1,…,𝑡 and decompose it into triangles using well-known construction via Latin square, that is, construct 𝑡×𝑡 Latin square and consider each element in the form (𝑎,𝑏,𝑐) where 𝑎 denotes the row, 𝑏 denotes the column, and 𝑐 denotes the entry with 1≤𝑎,𝑏,𝑐≤𝑡. Each cycle is of the form 𝑥_(1,𝑎),  𝑥_{(𝑟−1,𝑏)},  𝑥_(𝑟,𝑐). Then, for every triangle 𝑥_1,𝑎𝑥_{𝑟−1,𝑏}𝑥_𝑟,𝑐, replace the edge 𝑥_1,𝑎𝑥_{𝑟−1,𝑏} in each 𝐶_{(𝑟−1)𝑡}, by the edges 𝑥_{𝑟−1,𝑏}𝑥_𝑟,𝑐 and 𝑥_𝑟,𝑐𝑥_1,𝑎 to obtain cycles 𝐶_𝑟𝑡. Therefore, 𝐶_𝑟𝑡∣𝐶_𝑟∗𝐾_𝑡. Now take each cycle 𝐶_𝑟𝑡, make it into 𝐶_𝑟𝑡∗𝐾₂, and by Lemma 1, 𝐶_𝑟𝑡∗𝐾₂ has a decomposition into sunlet graphs 𝐿_2𝑟𝑡=𝐿_𝑞.
Subcase 2.2 (𝑚≡0(mod4)). Set 𝑚=2𝑡. The graph 𝐶_𝑟∗𝐾_𝑡 decomposes into Hamilton cycle 𝐶_𝑟𝑡 by [11]. Next, make each cycle 𝐶_𝑟𝑡 into 𝐶_𝑟𝑡∗𝐾₂. Each graph 𝐶_𝑟𝑡∗𝐾₂ decomposes into sunlet graph 𝐿_2𝑟𝑡 by Lemma 1.

Theorem 3. Let 𝑟,  𝑚 be positive integers satisfying 𝑟,𝑚≡0(mod4), then 𝐿_𝑟 decomposes 𝐶_𝑟∗𝐾_𝑚.

Proof. Let the partite sets (layers) of the 𝑟-partite graph 𝐶_𝑟∗𝐾_𝑚 be 𝑈₁,𝑈₂,…,𝑈_𝑟. Set 𝑚=2𝑡. Obtain a new graph from 𝐶_𝑟∗𝐾_𝑚 as follows.
Identify the subsets of vertices {𝑥_𝑖,𝑗}, for 1≤𝑖≤𝑟 and 1≤𝑗≤𝑚/2 into new vertices 𝑥¹_𝑖, and identify the subset of vertices {𝑥_𝑖,𝑗} for 1≤𝑖≤𝑟 and 𝑚/2+1≤𝑗≤𝑚 into new vertices 𝑥²_𝑖 and two of these vertices 𝑥^𝑘_𝑖, where 𝑘=1,2, are adjacent if and only if the corresponding subsets of vertices in 𝐶_𝑟∗𝐾_𝑚 induce 𝐾_𝑡,𝑡. The resulting graph is isomorphic to 𝐶_𝑟∗𝐾₂. Next, decompose 𝐶_𝑟∗𝐾₂ into cycles 𝐶_𝑟/2 as follows: 𝑥_𝑘,1𝑥_𝑘+1,1,…,𝑥_𝑑,1𝑥_𝑑−1,2,…,𝑥_𝑘+1,2,𝑥_𝑘,1𝑟𝑘=1,4𝑟+1,2+1,3𝑟4𝑟+1,…,𝑟−4𝑟+1,𝑑=4+𝑘, (4) where 𝑘,  𝑑 are calculated modulo 𝑟.
To construct the remaining cycles, apply mapping 𝜙 defined on the vertices.
Subcase 1.1 (𝑖 odd in each cycle). Consider 𝜙𝑥_𝑖,𝑗=𝑥_𝑖,𝑗+1. (5) This is the desired decomposition of 𝐶_𝑟∗𝐾₂ into cycles 𝐶_𝑟/2.
Subcase 1.2 (𝑖 even in each cycle). Consider 𝜙𝑥_𝑖,𝑗=𝑥_𝑖,𝑗. (6) This is the desired decomposition of 𝐶_𝑟∗𝐾₂ into cycles 𝐶_𝑟/2.
By lifting back these cycles 𝐶_𝑟/2 of 𝐶_𝑟∗𝐾₂ to 𝐶_𝑟∗𝐾_2𝑡, we get edge-disjoint subgraphs isomorphic to 𝐶_𝑟/2∗𝐾_𝑡. Obtain a new graph again from 𝐶_𝑟/2∗𝐾_𝑡 as follows.
For each 𝑗,1≤𝑗≤𝑡/2, identify the subsets of vertices {𝑥_{𝑖,2𝑗−1},𝑥_𝑖,2𝑗}, where 1≤𝑖≤𝑟/2 into new vertices 𝑥^𝑗_𝑖, and two of these vertices 𝑥^𝑗_𝑖 are adjacent if and only if the corresponding subsets of vertices in 𝐶_𝑟/2∗𝐾_𝑡 induce 𝐾_2,2. The resulting graph is isomorphic to 𝐶_𝑟/2∗𝐾_𝑡/2. Then, decompose 𝐶_𝑟/2∗𝐾_𝑡/2 into cycles 𝐶_𝑟/2. Each 𝐶_𝑟/2∗𝐾_𝑡/2 decomposes into cycles 𝐶_𝑟/2 by [12]. By lifting back these cycles 𝐶_𝑟/2 of 𝐶_𝑟/2∗𝐾_𝑡/2 to 𝐶_𝑟/2∗𝐾_𝑡, we get edge-disjoint subgraph isomorphic to 𝐶_𝑟/2∗𝐾₂. Finally, each 𝐶_𝑟/2∗𝐾₂ decomposes into two sunlet graphs 𝐿_𝑟 (by Lemma 1), and we have 𝐶_𝑟∗𝐾_𝑚 decomposing into sunlet graphs 𝐿_𝑟 as required.

Theorem 4 (see [12]). The cycle 𝐶_𝑚 decomposes 𝐶_𝑘∗𝐾_𝑚 for every even 𝑚>3.

Theorem 5 (see [12]). If 𝑚 and 𝑘≥3 are odd integers, then 𝐶_𝑚 decomposes 𝐶_𝑘∗𝐾_𝑚.

Theorem 6. The sunlet graph 𝐿_𝑚 decomposes 𝐶_𝑟∗𝐾_𝑚 if and only if either one of the following conditions is satisfied. (1)𝑟is a positive odd integer, and 𝑚 is a positive even integer. (2)𝑟,  𝑚 are positive even integers with 𝑚≡0(mod4).

Proof. (1) Set 𝑚=2𝑡, where 𝑡 is a positive integer. Let the partite sets (layers) of the 𝑟-partite graph 𝐶_𝑟∗𝐾_𝑚 be 𝑈₁,𝑈₂,…,𝑈_𝑟. For each 𝑗, where 1≤𝑗≤𝑡, identify the subsets of vertices {𝑥_{𝑖,2𝑗−1},𝑥_𝑖,2𝑗}, for 1≤𝑖≤𝑟 into new vertices 𝑥^𝑗_𝑖, and two of these vertices 𝑥^𝑗_𝑖 are adjacent if and only if the corresponding subsets of vertices in 𝐶_𝑟∗𝐾_𝑚 induce 𝐾_2,2. The resulting graph is isomorphic to 𝐶_𝑟∗𝐾_𝑡. Then, decompose 𝐶_𝑟∗𝐾_𝑡 into cycles 𝐶_𝑡, where 𝑡 is a positive integer.
Now, 𝐶_𝑡∣𝐶_𝑟∗𝐾_𝑡 by Theorems 4 and 5.
By lifting back these 𝑡-cycles of 𝐶_𝑟∗𝐾_𝑡 to 𝐶_𝑟∗𝐾_2𝑡, we get edge-disjoint subgraphs isomorphic to 𝐶_𝑡∗𝐾₂. Each copy of 𝐶_𝑡∗𝐾₂ decomposes into sunlet graphs of length 2𝑡 (by Lemma 1), and we have 𝐶_𝑟∗𝐾_𝑚 decomposing into sunlet graphs of length 𝑚 as required.
(2) Set 𝑚=2𝑡, where 𝑡 is an even integer since 𝑚≡0(mod4).
Obtain a new graph 𝐶_𝑟∗𝐾_𝑡 from the graph 𝐶_𝑟∗𝐾_𝑚 as in Case 1. By Theorem 4, 𝐶_𝑡∣𝐶_𝑟∗𝐾_𝑡. By lifting back these 𝑡-cycles of 𝐶_𝑟∗𝐾_𝑡 to 𝐶_𝑟∗𝐾_2𝑡, we get edge-disjoint subgraphs isomorphic to 𝐶_𝑡∗𝐾₂. Each copy of 𝐶_𝑡∗𝐾₂ decomposes into sunlet graph of length 2𝑡 (by Lemma 1). Therefore, 𝐿_𝑚∣𝐶_𝑟∗𝐾_𝑚 as required.

Remark 7. In [10], it was shown that 𝐿_2𝑟∗𝐾_𝑙canbedecomposedinto𝑙²copiesof𝐿_2𝑟. (7) This, coupled with Lemma 1, gives the following.

Theorem 8 (see [10]). The graph 𝐶_𝑟∗𝐾_2𝑙 decomposes into sunlet graphs 𝐿_2𝑟 for any positive integer 𝑙.

Lemma 9 (see [3]). Let 𝑛≥4 be an even integer. Then, 𝐾_𝑛+𝐼 is 𝐶_𝑛-decomposable.

Lemma 10 (see [3]). Let 𝑚 and 𝑛 be integers with 𝑚 odd, 𝑛≡2(mod4),  3≤𝑚≤𝑛<2𝑚, and 𝑛²≡0(mod2𝑚). Then, 𝐾_𝑛+𝐼 is 𝐶_𝑚-decomposable.

Lemma 11 (see [3]). Let 𝑚 and 𝑛 be integers with 𝑚 odd, 𝑛≡0(mod4),  3≤𝑚≤𝑛<2𝑚, and 𝑛²≡0(mod2𝑚). Then, 𝐾_𝑛+𝐼 is 𝐶_𝑚-decomposable.

We can now prove the major result.

Theorem 12. For any even integers 𝑚≥2,  𝑛>2, and 𝑞≥6, the sunlet graph 𝐿_𝑞 decomposes 𝐾_𝑛+𝐼∗𝐾_𝑚 if and only if 𝑛²𝑚²/2≡0(mod𝑞).

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 (𝑞∣𝑛)
Subcase 1.1 (𝑛>𝑞). Cycle 𝐶_𝑛 decomposes 𝐾_𝑛+𝐼 by Lemma 9, and we have 𝐶_𝑛∗𝐾_𝑚∣𝐾_𝑛+𝐼∗𝐾_𝑚. (8) Each graph 𝐶_𝑛∗𝐾_𝑚 decomposes into sunlet graph 𝐿_𝑞, where 𝑞=𝑛𝑚 by Lemma 2, and we have 𝐾_𝑛+𝐼∗𝐾_𝑚 decomposing into sunlet graph 𝐿_𝑞, where 𝑞>𝑛.
Subcase 1.2 (𝑞=𝑛). First, consider 𝑛≡0(mod4).
Cycle 𝐶_𝑞 decomposes 𝐾_𝑞+𝐼 by Lemma 9, and we have 𝐶_𝑞∗𝐾_𝑚∣𝐾_𝑞+𝐼∗𝐾_𝑚. (9) Now, sunlet graph 𝐿_𝑞∣(𝐶_𝑞∗𝐾_𝑚) by Theorem 3, and hence sunlet graph 𝐿_𝑞 decomposes 𝐾_𝑛+𝐼∗𝐾_𝑚.
Also, consider 𝑛≡2(mod4).
Suppose 𝑚=2𝑡. Cycle 𝐶_𝑞/2 decomposes 𝐾_𝑞+𝐼 by Lemma 10, and we have 𝐶_𝑞/2∗𝐾_2𝑡∣𝐾_𝑞+𝐼∗𝐾_2𝑡. (10) Now, sunlet graph 𝐿_𝑞 decomposes 𝐶_𝑞/2∗𝐾_2𝑡 by Theorem 8, and we have 𝐾_𝑛+𝐼∗𝐾_𝑚 decomposing into sunlet graph of length 𝑞.
Case 2 (𝑞∣𝑚)
Subcase 2.1 (𝑚≡0(mod4)). Suppose 𝑚=𝑞, and by Lemma 9, cycle 𝐶_𝑛 decomposes 𝐾_𝑛+𝐼, and we have 𝐶_𝑛∗𝐾_𝑞∣𝐾_𝑛+𝐼∗𝐾_𝑞. (11) Also, sunlet graph 𝐿_𝑞 decomposes each 𝐶_𝑛∗𝐾_𝑞 by Theorem 6, and we have sunlet graph 𝐿_𝑞 decomposing 𝐾_𝑛+𝐼∗𝐾_𝑚.
Subcase 2.2 (𝑚≡2(mod4)). Let 𝑚=𝑞 and 𝑟≤𝑛 an odd integer. Cycle 𝐶_𝑟 decomposes 𝐾_𝑛+𝐼, by Lemmas 9, 10, and 11, and we have 𝐶_𝑟∗𝐾_𝑞∣𝐾_𝑛+𝐼∗𝐾_𝑞. (12) Now, each 𝐶_𝑟∗𝐾_𝑞 decomposes into sunlet graph 𝐿_𝑞 by Theorem 6, and we have 𝐾_𝑛+𝐼∗𝐾_𝑚 decomposing into sunlet graph 𝐿_𝑞 as required.
Subcase 2.3 (𝑚>𝑞). Set 𝑚=𝑤𝑞, where 𝑤 is any positive integer, then by Subcases 2.1 and 2.2, we have 𝐿_𝑞∗𝐾_𝑤∣𝐾_𝑛+𝐼∗𝐾_𝑞∗𝐾_𝑤. (13) Each graph 𝐿_𝑞∗𝐾_𝑤 decomposes into sunlet graph 𝐿_𝑞 by Remark 7, and we have 𝐾_𝑛+𝐼∗𝐾_𝑚 decomposing into sunlet graph 𝐿_𝑞.