Let L2n 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 Cn. The necessary condition for the equipartite graph Kn+I*K̅m to be decomposed into L2n for n≥2 is that the order of L2n must divide n2m2/2, the order of Kn+I*K̅m. In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques.

1. Introduction

Let Cr, Kn, K-m denote cycle of length r, complete graph on n vertices, and complement of complete graph on m vertices. For n even, Kn+I denotes the multigraph obtained by adding the edges of a 1-factor to Kn, thus duplicating n/2 edges. The total number of edges in Kn+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 Cn⊙K1, that is, a cycle with pendant points which has 2n vertices. This is called sunlet graph and denoted by Lq, 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 Kn, the complete graph minus a 1-factor Kn-I [1, 2], and the complete graph plus a 1-factor Kn+I [3].

The study of cycle decomposition of Kn*K-m was initiated by Hoffman et al. [4]. The necessary and sufficient conditions for the existence of a Cp-decomposition of Kn*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 C2p-decomposition of Kn*K-m were given by Smith [7]. For a prime number p≥3, Smith [8] showed that C3p-decomposition of Kn*K-m exists if the obvious necessary conditions are satisfied. In [9], Anitha and Lekshmi proved that the complete graph Kn and the complete bipartite graph Kn,n for n even have decompositions into sunlet graph Ln. Similarly, in [10], it was shown that the complete equipartite graph Kn*K-m has a decomposition into sunlet graph of length 2p, for a prime p.

We extend these results by considering the decomposition of Kn+I*K-m into sunlet graphs and prove the following result.

Let m≥2, n>2, and q≥6 be even integers. The graph Kn+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 Kn+I*K-m.

2. Proof of the Result

To prove the result, we need the following.

Lemma 1 (see [<xref ref-type="bibr" rid="B3">10</xref>]).

For r≥3, L2r decomposes Cr*K-2.

Lemma 2.

For any integer r>2 and a positive even integer m, the graph Cr*K-m has a decomposition into sunlet graph Lq, for q=rm.

Proof

Case 1 (r is even). First observe that Cr*K-2 can be decomposed into 2 sunlet graphs with 2r vertices. Now, set m=2t and decompose Cr*K-t into cycles Crt. To decompose Cr*K-t into t-cycles Crt, denote vertices in ith part of Cr*K-t by xi,j for j=1,…,t, i=1,2,…,r and create t base cycles x1,jx2,jx3,j⋯xr-1,jxr,j. Next, combine these base cycles into one cycle Crt by replacing each edge x1,jx2,j with x1,jx2,j+1. To create the remaining cycles Crt, we apply mappings ϕs for s=0,1,…,t-1 defined on the vertices as follows.

Subcase 1.1 (i odd). Consider
(1)ϕs(xi,j)=xi,j.
This is the desired decomposition into cycles Crt.

Subcase 1.2 (i even). Consider
(2)ϕs(xi,j)=xi,j+s.
This is the desired decomposition into cycles Crt.

Now take each cycle Crt, and make it back into Crt*K-2. Each Crt*K-2 decomposes into 2 sunlet graphs L2rt (by Lemma 1), and we have Cr*K-m decomposing into sunlet graphs with length rm for r even. Note that
(3)Cr*K-2t=(Cr*K-t)*K-2.Case 2 (r is odd)

Subcase 2.1 (m≡2(mod4)). Set m=2t. First create t cycles C(r-1)t in Cr-1*K-t as in Case 1. Then, take complete tripartite graph Kt,t,t with partite sets Xi={xi,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 x1,axr-1,bxr,c, replace the edge x1,axr-1,b in each C(r-1)t, by the edges xr-1,bxr,c and xr,cx1,a to obtain cycles Crt. Therefore, Crt∣Cr*K-t. Now take each cycle Crt, make it into Crt*K-2, and by Lemma 1, Crt*K-2 has a decomposition into sunlet graphs L2rt=Lq.

Subcase 2.2 (m≡0(mod4)). Set m=2t. The graph Cr*K-t decomposes into Hamilton cycle Crt by [11]. Next, make each cycle Crt into Crt*K-2. Each graph Crt*K-2 decomposes into sunlet graph L2rt by Lemma 1.

Theorem 3.

Let r, m be positive integers satisfying r,m≡0(mod4), then Lr decomposes Cr*K-m.

Proof.

Let the partite sets (layers) of the r-partite graph Cr*K-m be U1,U2,…,Ur. Set m=2t. Obtain a new graph from Cr*K-m as follows.

Identify the subsets of vertices {xi,j}, for 1≤i≤r and 1≤j≤m/2 into new vertices xi1, and identify the subset of vertices {xi,j} for 1≤i≤r and m/2+1≤j≤m into new vertices xi2 and two of these vertices xik, where k=1,2, are adjacent if and only if the corresponding subsets of vertices in Cr*K-m induce Kt,t. The resulting graph is isomorphic to Cr*K-2. Next, decompose Cr*K-2 into cycles Cr/2 as follows: xk,1xk+1,1,…,xd,1xd-1,2,…,xk+1,2,xk,1(4)k=1,r4+1,r2+1,3r4+1,…,r-r4+1,d=r4+k,
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
(5)ϕ(xi,j)=xi,j+1.
This is the desired decomposition of Cr*K-2 into cycles Cr/2.

Subcase 1.2 (i even in each cycle). Consider
(6)ϕ(xi,j)=xi,j.
This is the desired decomposition of Cr*K-2 into cycles Cr/2.

By lifting back these cycles Cr/2 of Cr*K-2 to Cr*K-2t, we get edge-disjoint subgraphs isomorphic to Cr/2*K-t. Obtain a new graph again from Cr/2*K-t as follows.

For each j,1≤j≤t/2, identify the subsets of vertices {xi,2j-1,xi,2j}, where 1≤i≤r/2 into new vertices xij, and two of these vertices xij are adjacent if and only if the corresponding subsets of vertices in Cr/2*K-t induce K2,2. The resulting graph is isomorphic to Cr/2*K-t/2. Then, decompose Cr/2*K-t/2 into cycles Cr/2. Each Cr/2*K-t/2 decomposes into cycles Cr/2 by [12]. By lifting back these cycles Cr/2 of Cr/2*K-t/2 to Cr/2*K-t, we get edge-disjoint subgraph isomorphic to Cr/2*K-2. Finally, each Cr/2*K-2 decomposes into two sunlet graphs Lr (by Lemma 1), and we have Cr*K-m decomposing into sunlet graphs Lr as required.

Theorem 4 (see [<xref ref-type="bibr" rid="B6">12</xref>]).

The cycle Cm decomposes Ck*K-m for every even m>3.

Theorem 5 (see [<xref ref-type="bibr" rid="B6">12</xref>]).

If m and k≥3 are odd integers, then Cm decomposes Ck*K-m.

Theorem 6.

The sunlet graph Lm decomposes Cr*K-m if and only if either one of the following conditions is satisfied.

ris a positive odd integer, and m is a positive even integer.

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 Cr*K-m be U1,U2,…,Ur. For each j, where 1≤j≤t, identify the subsets of vertices {xi,2j-1,xi,2j}, for 1≤i≤r into new vertices xij, and two of these vertices xij are adjacent if and only if the corresponding subsets of vertices in Cr*K-m induce K2,2. The resulting graph is isomorphic to Cr*K-t. Then, decompose Cr*K-t into cycles Ct, where t is a positive integer.

Now, Ct∣Cr*K-t by Theorems 4 and 5.

By lifting back these t-cycles of Cr*K-t to Cr*K-2t, we get edge-disjoint subgraphs isomorphic to Ct*K-2. Each copy of Ct*K-2 decomposes into sunlet graphs of length 2t (by Lemma 1), and we have Cr*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 Cr*K-t from the graph Cr*K-m as in Case 1. By Theorem 4, Ct∣Cr*K-t. By lifting back these t-cycles of Cr*K-t to Cr*K-2t, we get edge-disjoint subgraphs isomorphic to Ct*K-2. Each copy of Ct*K-2 decomposes into sunlet graph of length 2t (by Lemma 1). Therefore, Lm∣Cr*K-m as required.

Remark 7.

In [10], it was shown that
(7)L2r*K-lcanbedecomposedintol2copiesofL2r.
This, coupled with Lemma 1, gives the following.

Theorem 8 (see [<xref ref-type="bibr" rid="B3">10</xref>]).

The graph Cr*K-2l decomposes into sunlet graphs L2r for any positive integer l.

Lemma 9 (see [<xref ref-type="bibr" rid="B13">3</xref>]).

Let n≥4 be an even integer. Then, Kn+I is Cn-decomposable.

Lemma 10 (see [<xref ref-type="bibr" rid="B13">3</xref>]).

Let m and n be integers with m odd, n≡2(mod4), 3≤m≤n<2m, and n2≡0(mod2m). Then, Kn+I is Cm-decomposable.

Lemma 11 (see [<xref ref-type="bibr" rid="B13">3</xref>]).

Let m and n be integers with m odd, n≡0(mod4), 3≤m≤n<2m, and n2≡0(mod2m). Then, Kn+I is Cm-decomposable.

We can now prove the major result.

Theorem 12.

For any even integers m≥2, n>2, and q≥6, the sunlet graph Lq decomposes Kn+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 Cn decomposes Kn+I by Lemma 9, and we have
(8)Cn*K-m∣Kn+I*K-m.
Each graph Cn*K-m decomposes into sunlet graph Lq, where q=nm by Lemma 2, and we have Kn+I*K-m decomposing into sunlet graph Lq, where q>n.

Subcase 1.2 (q=n). First, consider n≡0(mod4).

Cycle Cq decomposes Kq+I by Lemma 9, and we have
(9)Cq*K-m∣Kq+I*K-m.
Now, sunlet graph Lq∣(Cq*K-m) by Theorem 3, and hence sunlet graph Lq decomposes Kn+I*K-m.

Also, consider n≡2(mod4).

Suppose m=2t. Cycle Cq/2 decomposes Kq+I by Lemma 10, and we have
(10)Cq/2*K-2t∣Kq+I*K-2t.
Now, sunlet graph Lq decomposes Cq/2*K-2t by Theorem 8, and we have Kn+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 Cn decomposes Kn+I, and we have
(11)Cn*K-q∣Kn+I*K-q.
Also, sunlet graph Lq decomposes each Cn*K-q by Theorem 6, and we have sunlet graph Lq decomposing Kn+I*K-m.

Subcase 2.2 (m≡2(mod4)). Let m=q and r≤n an odd integer. Cycle Cr decomposes Kn+I, by Lemmas 9, 10, and 11, and we have
(12)Cr*K-q∣Kn+I*K-q.
Now, each Cr*K-q decomposes into sunlet graph Lq by Theorem 6, and we have Kn+I*K-m decomposing into sunlet graph Lq 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
(13)Lq*K-w∣(Kn+I*K-q)*K-w.
Each graph Lq*K-w decomposes into sunlet graph Lq by Remark 7, and we have Kn+I*K-m decomposing into sunlet graph Lq.

AlspachB.GavlasH.Cycle Decompositions of Kn and Kn-IŠajnaM.Cycle decompositions. III. Complete graphs and fixed length cyclesŠajnaM.Decomposition of the complete graph plus a 1-factor into cycles of equal lengthHoffmanD. G.LindnerC. C.RodgerC. A.On the construction of odd cycle systemsManikandanR. S.PaulrajaP.Cp-decompositions of some regular graphsManikandanR. S.PaulrajaP.C5-decompositions of the tensor product of complete graphsSmithB. R.Decomposing complete equipartite graphs into cycles of length 2pSmithB. R.Complete equipartite 3p-cycle systemsAnithaR.LekshmiR. S.N-sun decomposition of complete, complete bipartite and some Harary graphsAkwuA. D.AjayiD. O. A.Decomposing equipartite graphs into certain corona graphs of length 2pLaskarR.Decomposition of some composite graphs into Hamilton CyclesProceedings of the 5th Hungarian Colloquium keszthely1978North Holland705716FrončekD.KovářP.KubesaM.Decompositions of complete graphs into blown-up cycles C_{m}[2]