ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.867617 J. Algebra Comb. Discrete Appl. 8(1) • 41–51 Received: 31 March 2020 Accepted: 25 September 2020 Journal of Algebra Combinatorics Discrete Structures and Applications Decomposition of product graphs into sunlet graphs of order eight∗ Research Article Kaliappan Sowndhariya, Appu Muthusamy Abstract: For any integer k ≥ 3 , we define sunlet graph of order 2k, denoted by L2k, as the graph consisting of a cycle of length k together with k pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for the existence of L8-decomposition of tensor product and wreath product of complete graphs. 2010 MSC: 05C51 Keywords: Graph decomposition, Product graphs, Corona graph, Sunlet graph 1. Introduction All graphs considered here are finite, simple and undirected. For the standard graph-theoretic termi- nology the readers are referred to [7]. A cycle of length k is called k-cycle and it is denoted by Ck. Let Km denotes the complete graph on m vertices and Km,n denotes the complete bipartite graph with m and n vertices in the parts. We denote the complete m-partite graph with n1,n2, . . . ,nm vertices in the parts by Kn1,n2,...,nm. For any integer λ > 0, λG denotes the graph consisting of λ edge-disjoint copies of G. The complement of the graph G is denoted by G. The subgraph of G induced by S ⊆ V (G) is denoted as 〈S〉. For any two graphs G and H of orders m and n, respectively, the corona product G�H is the graph ob- tained by taking one copy of G, m copies of H and then joining the ith vertex of G to every vertex in the ith copy of H. We define the sunlet graph L2k with V (L2k) = {x1,x2, . . . ,xk,xk+1,xk+2, . . . ,x2k} and E(L2k) = {xixi+1,xixk+i | i = 1,2, ...,k and subscripts of the first term is taken addition modulo k}. We de- note it by L2k = ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) . Clearly, Ck �K1 ∼= L2k. For two graphs G and H, their tensor product G×H and lexicographic or wreath product G⊗H have ∗ This work was supported by Department of Science and Technology, University Grant Commission, Government of India. Kaliappan Sowndhariya(Corresponding Author), Appu Muthusamy; Department of Mathematics, Periyar Uni- versity, Salem, Tamil Nadu, India (email: sowndhariyak@gmail.com, ambdu@yahoo.com). 41 https://orcid.org/0000-0001-9014-6916 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 the same vertex set V (G)×V (H) = {(g,h) : g ∈ V (G) and h ∈ V (H)} and their edge sets are defined as follows: E(G×H) = {(g,h)(g′,h′) : gg′ ∈ E(G) and hh′ ∈ E(H)}, E(G ⊗ H) = {(g,h)(g′,h′) : gg′ ∈ E(G) or g = g′ and hh′ ∈ E(H)}. It is well known that the above products are associative and distributive over edge-disjoint unions of graphs and the tensor product is commutative. It is easy to observe that Km ⊗Kn ∼= Kn,n,...n(m times). We shall use the following notation throughout the paper. Let G and H be simple graphs with vertex sets V (G) = {x1,x2, . . . ,xn} and V (H) = {y1,y2, . . . ,ym}. Then for our convenience, we write V (G) × V (H) = ⋃n i=1 Xi, where Xi stands for xi × V (H). Further, in the sequel, we shall denote the vertices of Xi as { x j i|1 ≤ j ≤ m } , where xji stands for the vertex (xi,yj) ∈ V (G)×V (H). A labeling of a graph G with n edges is an injection ρ from V (G), the vertex set of G, into a subset S ⊆ Z2n+1, the additive group Z2n+1. The length of an edge e = xy with end vertices x and y is defined as l(xy) = min{ρ(x)−ρ(y),ρ(y)−ρ(x)}. Note that the subtraction is performed in Z2n+1 and hence 1 ≤ l(e) ≤ n. If the length of the n edges are distinct and is equal to {1,2, . . . ,n}, then ρ is a rosy labeling; moreover, if S ⊆ {1,2, . . . ,n}, then ρ is a graceful labeling. A graceful labeling is said to be an α-labeling if there exists a number α0 with the property that for every edge e = xy in G with α(x) < α(y) it holds that α(x) ≤ α0 < α(y). By a decomposition of a graph G, we mean a list of edge-disjoint subgraphs of G whose union is G. For a graph G, if E(G) can be partitioned into E1,E2, ...,Ek such that the subgraph induced by Ei is Hi, for all i, 1 ≤ i ≤ k, then we say that H1,H2, ...,Hk decompose G and we write G = H1 ⊕H2 ⊕ ...⊕Hk, since H1,H2, ...,Hk are edge-disjoint subgraphs of G. For 1 ≤ i ≤ k, if Hi ∼= H, we say that G has a H- decomposition. Study of H-decomposition of graphs is not new. Many authors around the world are working in the field of cycle decomposition [4, 8, 9, 21, 22], path decomposition [24, 25], star decompositon [19, 23, 26, 27] and Hamilton cycle decomposition [2, 3, 15, 16] problems in graphs. Here we consider the sunlet decompo- sition of product graphs. Anitha and Lekshmi [5, 6] proved that n-sun decomposition of complete graph, complete bipartite graph and the Harary graphs. Liang and Guo [17, 18] gave the existence spectrum of a k-sun system of order v as k = 2,4,5,6,8. Fu et. al. [12, 13] obtained that 3-sun decompositions of Kp,p,r, Knand embed a cyclic steiner triple system of order n into a 3-sun system of order 2n − 1, for n = 1(mod 6). Further they obtained k-sun system when k = 6,10,14,2t, for t > 1. Fu et. al. [11] obtained the existence of a 5-sun system of order v. Gionfriddo et.al. [14] obtained the spectrum for uniformly resolvable decompositions of Kv into 1-factor and h-suns. Akwu and Ajayi [1] obtained the necessary and sufficient conditions for the existence of decomposition of Kn ⊗Km and (Kn − I) ⊗Km, where I denote the 1-factor of a complete graph into sunlet graph of order twice the prime. In this paper, we obtain the decomposition of some product graphs into sunlet graphs of order eight which is the least even order not proved so far for product graphs, which motivate us to consider this prob- lem. In Section 2, we obtain the necessary and sufficient conditions for the existence of L8-decomposition of complete bipartite graphs with part size m and n. In Section 3, we obtain the necessary and sufficient conditions for the existence of L8-decomposition of tensor product of complete graphs. In Section 4, we obtain the necessary and sufficient conditions for the existence of L8-decomposition of complete multi- partite graphs with uniform part size. To prove our results, we state the following: Theorem 1.1. [20] For all n ≥ 3, Cn �K1 is an α-labeling. Theorem 1.2. [10] Let G be a graph with n edges. If G admits a rosy labeling, then it decomposes K2n+1; if G admits an α-labeling, then it decomposes K2np+1 for every p > 0. Theorem 1.3. [13] Let t ≥ 2 be an integer. An L2.2t-decomposition of Kn exists if and only if n ≡ 0 (or) 1 (mod 2t+2). Remark 1.4. If n ≡ 0 (mod 4), then K4,n can be decomposed as copies of K4,4 and L8- decomposition of K4,4 is shown in below figure. Therefore L8-decomposition exists in K4,n for n is a multiple of 4. 42 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Figure 1. L8- decomposition of K4,4. 2. L8- decomposition of Km,n Now we obtain the necessary and sufficient conditions for the existence of an L8-decomposition of Km,n as follows. Let the vertices of Km,n be {x1,x2, ...,xm,y1,y2, ...,yn}. Lemma 2.1. There exists an L8- decomposition of K8,6. Proof. We exhibit the L8- decomposition of K8,6 as follows:( x1 y1 x2 y2 y3 x5 y4 x6 ) , ( x3 y3 x4 y4 y5 x2 y6 x1 ) , ( x5 y5 x6 y6 y3 x1 y4 x2 ) , ( x7 y5 x8 y6 y1 x2 y2 x1 ) ,( x7 y3 x8 y4 y2 x6 y1 x5 ) , ( x3 y1 x4 y2 y6 x6 y5 x5 ) . Lemma 2.2. There exists an L8- decomposition of K8,7. Proof. We exhibit the L8- decomposition of K8,7 as follows:( x1 y1 x2 y2 y5 x4 y7 x7 ) , ( x3 y3 x4 y4 y2 x1 y5 x2 ) , ( x5 y5 x6 y6 y4 x3 y7 x1 ) , ( x7 y1 x8 y7 y3 x6 y6 x5 ) ,( x7 y4 x8 y5 y6 x1 y3 x2 ) , ( x3 y6 x4 y7 y1 x2 y2 x1 ) , ( x5 y2 x6 y3 y1 x8 y4 x2 ) . Lemma 2.3. There exists an L8- decomposition of K8,9. Proof. We exhibit the L8- decomposition of K8,9 as follows:( x1 y1 x2 y2 y3 x6 y4 x7 ) , ( x3 y3 x4 y4 y8 x5 y5 x6 ) , ( x5 y5 x6 y6 y1 x3 y7 x4 ) , ( x7 y7 x8 y8 y5 x4 y6 x5 ) ,( x3 y1 x4 y9 y6 x8 y2 x7 ) , ( x5 y2 x6 y9 y7 x3 y8 x8 ) , ( x7 y3 x8 y4 y1 x6 y2 x5 ) , ( x1 y5 x2 y6 y4 x8 y9 x7 ) ,( x1 y7 x2 y8 y9 x3 y3 x4 ) . Lemma 2.4. There exists an L8- decomposition of K12,6. Proof. We exhibit the L8- decomposition of K12,6 as follows:( x1 y1 x2 y2 y3 x12 y4 x11 ) , ( x3 y3 x4 y4 y5 x2 y6 x1 ) , ( x5 y5 x6 y6 y1 x1 y2 x2 ) , ( x7 y1 x8 y2 y3 x9 y4 x10 ) ,( x9 y3 x10 y4 y6 x11 y5 x12 ) , ( x11 y5 x12 y6 y1 x2 y2 x1 ) , ( x3 y1 x4 y2 y6 x10 y5 x9 ) , ( x5 y3 x6 y4 y2 x12 y1 x11 ) ,( x7 y5 x8 y6 y4 x9 y3 x10 ) . Lemma 2.5. There is no L8- decomposition of K8,5. 43 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Proof. Let A and B be the partite set of K8,5 such that |A| = 8 and |B| = 5. In L8, four vertices are of degree 3 and four vertices are of degree 1. Since K8,5 is a bipartite graph, then L8 has two vertices of degree 3 and two vertices of degree 1 in one partite set A and similarly in B. Total number of edges in K8,5 is 40, then we have 5L8 in K8,5. First we pull out 4L8 from K8,5(as shown in Fig.2). Since each vertices in A has degree 5, the remaining degree of each vertices of K8,5\4L8in the set A is 1. Here we can’t find a L8 in K8,5\4L8, since we need atleast two vertices of degree 3. Hence we conclude that L8-decomposition does not exists in K8,5. Figure 2. 4L8 in K8,5 Lemma 2.6. There is no L8- decomposition of K4,n for n ≡ 2 (mod 4). Proof. Let n = 4s + 2 for some s > 0. Suppose that K4,n has an L8-decomposition, then it has (2s + 1)L8. Let A = {x1,x2,x3,x4} and B = {y1,y2, ...,y4s+1,y4s+2} be the partite sets of K4,n. Consider the (2s)L8 = { L18,L 2 8, ...,L 2s 8 } which exists in K4,n−2. Now we have to find the last L8 i.e., L2s+18 . Let (x1y4s+1x2y4s+2) be a cycle in K4,n. Then join y4s+1 to x3 and y4s+2 to x4. Now we have to find pendant edges to the vertices x1 and x2. Suppose there are vertices ya and yb which are joined to x1 and x2, resp, as the pendant edges in L t1 8 for some t1 ∈{1,2, ...,2s}. Then we can join these edges to the vertices x1 and x2 in L 2s+1 8 . Suppose ya = y4s+1 and yb = y4s+2 or viceversa, then we can join the remaining edges x3y4s+2,x4y4s+1 in K4,n to ya and yb, resp. Therefore degLt18 (y4s+1) = degLt18 (y4s+2) = 3 and degL2s+18 (y4s+1) = degL2s+18 (y4s+2) = 3. This implies deg(y4s+1) = deg(y4s+2) = 6,which is a contradiction. Therefore ya 6= y4s+1 and yb 6= y4s+2 or viceversa. Then we find the pendant edges to ya and yb. There exist vertices xi and xj, i,j ∈{1,2,3,4} which are joined to ya and yb,resp, as the pendant edges in L t2 8 for some t1 6= t2 ∈ {1,2, ...,2s}. Then we can join these edges to the vertices ya and yb in L t1 8 . Now xi 6= x3 and xj 6= x4 or viceversa, since x3ya,x3yb,x4ya and x4yb are the edges of the cycle in L t1 8 . Then xi,xj must be x1,x2. Again we have to find the pendant edges to x1,x2. Repeat the above procedure cyclically we get to find the pendant edges of x1,x2. Therefore we can’t find the pendant edges to x1,x2. Hence the proof. Theorem 2.7. For any m,n ≥ 4, Km,n has an L8- decomposition if and only if mn ≡ 0 (mod 8) except (m,n) = (4,2 (mod 4)) & (8,5). Proof. Necessity. We first observe that Km,n has m + n vertices and mn edges. Assume that Km,n admits an L8- decomposition. Then the number of edges in the graph must be divisible by 8 i.e., 8|mn and hence mn ≡ 0 (mod 8). Further, (m,n) 6= (4,2 (mod 4)) & (8,5) follows from Lemmas 2.6 and 2.5. Sufficiency. We construct the required decomposition in two cases. Case(1) m (or) n ≡ 0 (mod 8). Suppose we take m ≡ 0 (mod 8). Further we divide the proof into four subcases. Subcase(i) m ≡ 0 (mod 8)and n ≡ 0 (mod 4). Let m = 8s and n = 4t for some s,t > 0. Then we can write Km,n = 2stK4,4. We know that K4,4 has an L8-decomposition(see Fig.2). 44 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Subcase(ii) m ≡ 0 (mod 8)and n ≡ 1 (mod 4). Let m = 8s and n = 4t + 1 for some s,t > 1, since for s = t = 1, Km,n has no L8- decomposition by Lemma 2.5. Then we can write Km,n = 2s(t − 2)K4,4 ⊕ sK8,9. Then by Lemma 2.3, we get an L8- decomposition of Km,n. Subcase(iii) m ≡ 0 (mod 8)and n ≡ 2 (mod 4). Let m = 8s and n = 4t + 2 for some s,t > 0. Then we can write Km,n = 2s(t−1)K4,4 ⊕sK8,6. Then by Lemma 2.1, we get an L8- decomposition of Km,n. Subcase(iv) m ≡ 0 (mod 8)and n ≡ 3 (mod 4). Let m = 8s and n = 4t + 3 for some s,t > 0. Then we can write Km,n = 2s(t−1)K4,4 ⊕sK8,7. Then by Lemma 2.2, we get an L8- decomposition of Km,n. Case(2) m ≡ 0 (mod 4)and n ≡ 0 (mod 2). Subcase(i) m ≡ 0 (mod 4)and n ≡ 0 (mod 4). Let m = 4s and n = 4t for some s,t > 0. Then we can write Km,n = stK4,4. We know that K4,4 has an L8-decomposition. Subcase(ii) m ≡ 0 (mod 4)and n ≡ 2 (mod 4). Let m = 4s and n = 4t + 2 for some s,t > 0. For s = 1, K4,n has no L8- decomposition by Lemma 2.6. Consider s ≥ 2. For even s, m must be the multiple of 8. Then by case(1), result is proved for even s. It is sufficient to prove the case for odd s. Consider s is odd and s ≥ 3. Then we can write Km,n = s(t− 1)K4,4 ⊕K4(s−3),6 ⊕K12,6. Since s is odd, s− 3 is even. Hence the results follows by the above cases and by the Lemma 2.4. 3. L8- decomposition of Km ×Kn In this section we investigate the existence of L8- decomposition of the tensor product of complete graphs. Lemma 3.1. For an even integer k > 2 and any graph G, there exists an L2k- decomposition of L2k×G. Proof. Let L2k be ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) and yj1yj2 be any edge in G, then the induced subgraph 〈L2k ×{yj1yj2}〉 of L2k ×G gives two L2k’s as follows:( x j1 1 x j2 2 x j1 3 . . . x j2 k x j2 k+1 x j1 k+2 x j2 k+3 . . . x j1 2k ) , ( x j2 1 x j1 2 x j2 3 . . . x j1 k x j1 k+1 x j2 k+2 x j1 k+3 . . . x j2 2k ) . So, for each edge in G there are two L2k’s in L2k ×G, and hence we have 2|E(G)| L2k’s in L2k ×G. Lemma 3.2. There exists an L8- decomposition of K4 ×K4. Proof. The L8- decomposition of K4 ×K4 is given below.( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1 4 x j2 1 x j1 2 ) for j1 < j2 ∈ {1,2,3,4}, ( x j+2 1 x j 2 x j+2 3 x j 4 x j+1 4 x j+1 1 x j+1 2 x j+1 3 ) for j = 1,2, ( x41 x 1 2 x 4 3 x 1 4 x32 x 2 3 x 3 4 x 2 1 ) . Lemma 3.3. There exists an L8- decomposition of K4 ×K5. Proof. The L8- decomposition of K4 ×K5 is given as follows:( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1 4 x j2 1 x j1 2 ) for j1 < j2 ∈{1,2,3,4,5} except (j1,j2) = (2,4),(3,4),( x31 x 4 2 x 3 3 x 4 4 x43 x 5 1 x 4 1 x 5 3 ) , ( x21 x 4 2 x 2 3 x 4 4 x43 x 5 3 x 4 1 x 5 1 ) , ( x31 x 1 2 x 3 3 x 1 4 x22 x 2 1 x 2 4 x 2 3 ) , ( x41 x 1 2 x 4 3 x 1 4 x32 x 2 3 x 3 4 x 2 1 ) ,( x41 x 2 2 x 4 3 x 2 4 x34 x 3 3 x 3 2 x 3 1 ) , ( x51 x 2 2 x 5 3 x 2 4 x12 x 4 4 x 1 4 x 4 2 ) , ( x51 x 3 2 x 5 3 x 3 4 x14 x 4 4 x 1 2 x 4 2 ) . Lemma 3.4. There exists an L8- decomposition of K4 ×K4,5. 45 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Proof. We can write K4,5 = K4,4 ⊕K4,1. Now K4 ×K4,5 = (K4 ×K4,4)⊕ (K4 ×K4,1). By Theorem 2.7 and Lemma 3.1, it is sufficient to prove the existence of L8- decomposition of K4 × K4,1. The L8-decomposition of K4 ×K4,1 shown in Fig.3 gives the required decomposition. Figure 3. L8 decomposition of K4 × K4,1. Lemma 3.5. There exists an L8- decomposition of K8 ×K7. Proof. We can write K8 = 3L8⊕C4 (see Fig.4). Now K8×K7 = 3 (L8 ×K7)⊕(C4 ×K7). To complete the proof, by Lemma 3.1, it is sufficient to prove the existence of L8- decomposition of C4 ×K7, which is given as follows:( x11 x 2 2 x 1 3 x 2 4 x74 x 7 3 x 7 2 x 7 1 ) , ( x21 x 1 2 x 2 3 x 1 4 x74 x 7 1 x 7 2 x 7 3 ) , ( x21 x 3 2 x 2 3 x 3 4 x54 x 6 1 x 5 2 x 6 3 ) , ( x31 x 2 2 x 3 3 x 2 4 x62 x 7 1 x 6 4 x 7 3 ) ,( x31 x 4 2 x 3 3 x 4 4 x64 x 7 1 x 6 2 x 7 3 ) , ( x41 x 3 2 x 4 3 x 3 4 x72 x 6 3 x 7 4 x 6 1 ) , ( x41 x 5 2 x 4 3 x 5 4 x74 x 7 3 x 7 2 x 7 1 ) , ( x51 x 4 2 x 5 3 x 4 4 x12 x 7 3 x 1 4 x 7 1 ) ,( x51 x 6 2 x 5 3 x 6 4 x14 x 1 1 x 1 2 x 1 3 ) , ( x61 x 5 2 x 6 3 x 5 4 x12 x 1 3 x 1 4 x 1 1 ) , ( x61 x 7 2 x 6 3 x 7 4 x14 x 1 1 x 1 2 x 1 3 ) , ( x71 x 6 2 x 7 3 x 6 4 x52 x 1 3 x 5 4 x 1 1 ) ,( x11 x 3 2 x 1 3 x 3 4 x52 x 7 1 x 5 4 x 7 3 ) , ( x31 x 1 2 x 3 3 x 1 4 x72 x 7 3 x 7 4 x 7 1 ) , ( x21 x 4 2 x 2 3 x 4 4 x62 x 1 1 x 6 4 x 1 3 ) , ( x41 x 2 2 x 4 3 x 2 4 x12 x 6 1 x 1 4 x 6 3 ) ,( x31 x 5 2 x 3 3 x 5 4 x74 x 2 1 x 7 2 x 2 3 ) , ( x51 x 3 2 x 5 3 x 3 4 x24 x 7 3 x 2 2 x 7 1 ) , ( x41 x 6 2 x 4 3 x 6 4 x14 x 2 3 x 1 2 x 2 1 ) , ( x61 x 4 2 x 6 3 x 4 4 x24 x 1 3 x 2 2 x 1 1 ) ,( x51 x 7 2 x 5 3 x 7 4 x22 x 2 1 x 2 4 x 2 3 ) . Figure 4. K8 = 3L8 ⊕ C4. Lemma 3.6. There exists an L8-decomposition of K8 ×K4,7. 46 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Proof. We can write K8 ×K4,7 = 2(K4 ×K4,4) ⊕ 2(K4 ×K4,3) ⊕ (K4,4 ×K4,4) ⊕ (K4,4 ×K4,3). By Theorem 2.7 and Lemma 3.1, it is sufficient to prove the existence of L8- decomposition of K4 × K4,3. The L8- decomposition of K4 ×K4,3 is given below.( x j1 1 x j2 2 x 1 3 x j2 4 x j2 3 x j1+1 4 x j2 1 x j1+1 2 ) , ( x j2 1 x j1 2 x j2 3 x j1 4 x j1+1 3 x j2 4 x j1+1 1 x j2 2 ) for j1 = 3, j2 ∈{5,6,7}( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1+2 3 x j2 1 x j1+2 1 ) , ( x j2 1 x j1 2 x j2 3 x j1 4 x j1+2 4 x j2 4 x j1+2 2 x j2 2 ) for j1 = 2, j2 ∈{5,6,7}( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1+3 1 x j2 1 x j1+3 3 ) , ( x j2 1 x j1 2 x j2 3 x j1 4 x j1+3 2 x j2 4 x j1+3 4 x j2 2 ) for j1 = 1, j2 ∈{5,6,7}. Lemma 3.7. There exists an L8-decomposition of K5 ×K5. Proof. We exhibit the L8- decomposition of K5 ×K5 as follows:( x11 x 2 3 x 1 4 x 2 5 x52 x 1 2 x 2 1 x 3 2 ) , ( x21 x 1 3 x 2 4 x 1 5 x12 x 2 2 x 4 2 x 5 3 ) , ( x21 x 3 3 x 2 4 x 3 5 x32 x 2 2 x 5 2 x 1 3 ) , ( x31 x 2 3 x 3 4 x 2 5 x12 x 3 2 x 5 2 x 4 2 ) ,( x31 x 4 3 x 3 4 x 4 5 x52 x 3 2 x 1 2 x 5 3 ) , ( x41 x 3 3 x 4 4 x 3 5 x32 x 4 2 x 5 2 x 5 3 ) , ( x41 x 5 3 x 4 4 x 5 5 x52 x 4 2 x 2 2 x 1 2 ) , ( x51 x 4 3 x 5 4 x 4 5 x42 x 5 2 x 1 2 x 3 2 ) ,( x11 x 5 3 x 1 4 x 5 5 x32 x 1 2 x 5 1 x 4 2 ) , ( x51 x 1 3 x 5 4 x 1 5 x12 x 5 2 x 4 1 x 4 2 ) , ( x11 x 3 3 x 1 4 x 3 5 x44 x 1 2 x 5 2 x 2 2 ) , ( x31 x 1 3 x 3 4 x 1 5 x14 x 3 2 x 2 1 x 2 2 ) ,( x31 x 5 3 x 3 4 x 5 5 x54 x 3 2 x 4 1 x 2 3 ) , ( x51 x 3 3 x 5 4 x 3 5 x32 x 5 5 x 2 1 x 5 2 ) , ( x21 x 4 3 x 2 4 x 4 5 x42 x 2 2 x 5 1 x 5 2 ) , ( x41 x 2 3 x 4 4 x 2 5 x12 x 4 2 x 5 1 x 5 2 ) ,( x11 x 4 3 x 1 4 x 4 5 x54 x 1 2 x 4 1 x 2 2 ) , ( x41 x 1 3 x 4 4 x 1 5 x22 x 4 2 x 2 1 x 5 2 ) , ( x21 x 5 3 x 2 4 x 5 5 x52 x 2 2 x 4 1 x 3 2 ) , ( x51 x 2 3 x 5 4 x 2 5 x22 x 5 2 x 3 2 x 4 3 ) ,( x12 x 2 4 x 3 2 x 4 4 x25 x 1 1 x 1 5 x 3 1 ) , ( x14 x 2 2 x 3 4 x 4 2 x32 x 5 5 x 5 1 x 5 4 ) , ( x13 x 2 5 x 3 3 x 4 5 x55 x 5 3 x 5 2 x 1 2 ) , ( x15 x 2 3 x 3 5 x 4 3 x33 x 4 5 x 1 2 x 5 5 ) ,( x11 x 2 2 x 3 1 x 4 2 x34 x 5 4 x 2 4 x 3 5 ) . Theorem 3.8. Km ×Kn has an L8- decomposition if and only if mn(m−1)(n−1) ≡ 0 (mod 16). Proof. Necessity. Assume that Km ×Kn admits an L8- decomposition. Then the number of edges in the graph Km × Kn is mn(m−1)(n−1) 2 which should be divisible by 8, the number of edges in L8 i.e., 16|mn(m−1)(n−1) and hence mn(m−1)(n−1) ≡ 0 (mod 16). Sufficiency. We construct the required decomposition in the following cases. Case(1) m,n ≡ 0 (mod 4). Let m = 4s and n = 4t for some s,t > 0. Then we can write Km × Kn = st(K4 ×K4) ⊕ 2st(s − 1)(t − 1)K4,16 ⊕ 2st(s + t − 2)K4,12. By Lemma 3.2 and Theorem 2.7 the graph Km × Kn has the desired decomposition. Case(2) m ≡ 0 (mod 4), n ≡ 1 (mod 4). Let m = 4s and n = 4t + 1 for some s,t > 0. Then we can write Km × Kn = (K4s × K4(t−1)) ⊕ s(K4 ×K5) ⊕ ( 5s(s−1) 2 ) K4,16 ⊕ s(t− 1) (K4 ×K4,5) ⊕ 4s(s− 1)(t− 1)K4,20. By the Case (1) above, Lemmas 3.3, 3.4 and Theorem 2.7 the graph Km ×Kn has the desired decomposition. Case(3) m ≡ 0 (mod 8), n ≡ 3 (mod 4). Subcase(i) m = 8 and n ≡ 3 (mod 4) Let n = 4t + 3 for some t > 0. Then we can write K8 × Kn = ( K8 ×K4(t−1) ) ⊕ (K8 ×K7) ⊕ (t−1) (K8 ×K4,7). The L8- decomposition of all the three terms follows from Case (1) and the Lemmas 3.5, 3.6. Subcase(ii) m ≡ 0 (mod 8), m > 8 and n ≡ 3 (mod 4) Now, let m = 8s for some s > 1. Then we can write Km×Kn = s(K8 ×Kn) ⊕ ( s(s−1) 2 ) (K8,8 ×Kn). Km ×Kn has the desired decomposition, by the Theorem 2.7, Lemma 3.1 and Subcase 3(i) above. Case(4) m ≡ 0 (mod 16). 47 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Let m = 16s, for some s > 0. We can write Km ×Kn = s(K16 ×Kn) ⊕ ( s(s−1) 2 ) (K16,16 ×Kn). The L8- decomposition of the terms in RHS follows from Lemma 3.1 and Theorems 1.3, 2.7. Case(5) m ≡ 1 (mod 16). Let m = 16s + 1, for some s > 0. Then by Theorem 1.3, we have L8- decomposition of K16s+1, for any s > 0. Then by Lemma 3.1, Km ×Kn has an L8- decomposition. Case(6) m ≡ 1 (mod 4), n ≡ 1 (mod 4). Let m = 4s + 1 and n = 4t + 1 for some s,t > 0. Then we can write Km ×Kn = ( K4(s−1) ×K4(t−1) ) ⊕ ( K4(s−1) ×K5 ) ⊕ ( K5 ×K4(t−1) ) ⊕ 2(s− 1)(t− 1) (K4 ×K4,5) ⊕ 4(s− 1)(t− 1)(s + t− 4)K4,20 ⊕ K5 ×K5 ⊕ 5 (s + t−2)K4,20 ⊕ 2(s−1)(t−1)K16,25. Then by the Cases (1), (2) above and by Lemmas 3.4, 3.7 and Theorem 2.7, the graph Km ×Kn has the desired decomposition. 4. L8- decomposition of Km ⊗Kn In this section we investigate the existence of L8- decomposition of wreath product of complete graphs. Lemma 4.1. If the graph G has an L2k-decomposition, then G⊗Kn has an L2k-decomposition for any n > 0 and even k > 2. Proof. Let G has an L2k- decomposition. For each L2k, ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) , in G, we exhibit the L2k- decomposition of L2k ⊗ Kn as follows: ( x j1 1 x j2 2 . . . x j2 k x j2 k+1 x j1 k+2 . . . x j1 2k ) for j1 ≤ j2 ∈ {1,2, ...,n},( x j2 1 x j1 2 . . . x j1 k x j1 k+1 x j2 k+2 . . . x j2 2k ) for j1 < j2 ∈{1,2, ...,n}. Lemma 4.2. There exists an L8- decomposition of K4,5 ⊗K6. Proof. The L8- decomposition of K4,5 ⊗K6 is given as follows:( x j1 1 x j2 5 x j1 2 x j2 6 x j2 7 x j1 3 x j2 8 x j1 4 ) for j1 ≤ j2 ∈{1,2,3,4,5,6} except (j1,j2) = (4,4),(2,3), (2,4),(3,4),( x j2 1 x j1 5 x j2 2 x j1 6 x j1 7 x j2 3 x j1 8 x j2 4 ) for j1 < j2 ∈{1,2,3,4,5,6} except (j1,j2) = (3,4), (3,5),(3,6),(4,5),(4,6),( x j1 3 x j2 7 x j1 4 x j2 8 x j2 6 x j1 2 x j2 5 x j1 1 ) for j1 ≤ j2 ∈{1,2,3,4,5,6} except (j1,j2) = (2,4),(2,5),( x j2 3 x j1 7 x j2 4 x j1 8 x j1 6 x j2 2 x j1 5 x j2 1 ) for j1 < j2 ∈{1,2,3,4,5,6} except (j1,j2) = (3,5),(4,5),( x2i1 x 1 i2 x4i1 x 3 i2 x5i2 x 3 i1 x4i2 x 1 i1 ) , ( x1i1 x 2 i2 x3i1 x 4 i2 x5i2 x 6 i1 x3i2 x 2 i1 ) , ( x4i1 x 2 i2 x5i1 x 6 i2 x5i2 x 2 i1 x3i2 x 6 i1 ) for i1 ∈{1,2} & i2 ∈{9},( x5i1 x 1 i2 x6i1 x 5 i2 x4i2 x 1 i1 x3i2 x 3 i1 ) for i1 ∈{1,2,3,4} & i2 ∈{9},( x41 x 4 5 x 4 2 x 4 6 x47 x 4 3 x 4 8 x 5 3 ) , ( x21 x 6 9 x 2 2 x 3 6 x37 x 1 2 x 3 8 x 2 4 ) , ( x21 x 4 5 x 2 2 x 4 6 x47 x 2 3 x 4 8 x 3 4 ) , ( x31 x 4 5 x 3 2 x 6 9 x47 x 3 3 x 4 8 x 1 1 ) ,( x41 x 3 5 x 4 2 x 3 6 x37 x 2 1 x 3 8 x 4 4 ) , ( x51 x 3 5 x 5 2 x 3 6 x37 x 2 2 x 3 8 x 5 4 ) , ( x61 x 3 5 x 6 2 x 3 6 x37 x 5 4 x 3 8 x 6 4 ) , ( x51 x 4 5 x 5 2 x 4 6 x47 x 5 3 x 4 8 x 3 1 ) ,( x61 x 4 5 x 6 2 x 4 6 x47 x 6 3 x 4 8 x 3 2 ) , ( x23 x 4 7 x 2 4 x 4 8 x35 x 2 2 x 4 5 x 2 1 ) , ( x23 x 5 7 x 2 4 x 5 8 x56 x 2 2 x 4 6 x 2 1 ) , ( x53 x 3 7 x 5 4 x 3 8 x36 x 5 2 x 4 6 x 5 1 ) , 48 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 ( x53 x 4 7 x 5 4 x 4 8 x35 x 5 2 x 4 5 x 5 1 ) , ( x23 x 1 9 x 4 3 x 3 9 x59 x 3 3 x 3 5 x 1 3 ) , ( x24 x 1 9 x 4 4 x 3 9 x59 x 3 4 x 4 6 x 1 4 ) , ( x13 x 2 9 x 3 3 x 4 9 x59 x 6 3 x 3 9 x 6 1 ) ,( x14 x 2 9 x 3 4 x 4 9 x59 x 6 4 x 3 9 x 6 2 ) , ( x43 x 2 9 x 5 3 x 6 9 x59 x 2 3 x 3 9 x 3 3 ) , ( x44 x 2 9 x 5 4 x 6 9 x59 x 2 4 x 3 9 x 3 4 ) , ( x23 x 4 9 x 6 3 x 6 9 x46 x 4 3 x 3 5 x 1 3 ) ,( x24 x 4 9 x 6 4 x 6 9 x55 x 4 4 x 4 6 x 1 4 ) . Lemma 4.3. There exists an L8-decomposition of K4,5 ⊗K10. Proof. We exhibit the L8- decomposition of K4,5 ⊗K10 as follows:( x1i1 x 1 i2 x2i1 x 2 i2 x7i2 x 5 i1 x8i2 x 6 i1 ) , ( x1i1 x 3 i2 x2i1 x 4 i2 x8i2 x 6 i1 x7i2 x 5 i1 ) , for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9},( x3i1 x 1 i2 x4i1 x 2 i2 x9i2 x 7 i1 x10i2 x 8 i1 ) , ( x1i1 x 5 i2 x2i1 x 6 i2 x9i2 x 9 i1 x10i2 x 10 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9},( x3i1 x 5 i2 x4i1 x 6 i2 x8i2 x 8 i1 x7i2 x 7 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9},( x9i1 x 1 i2 x10i1 x 2 i2 x3i2 x 6 i1 x4i2 x 5 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (1,9)( x3i1 x 3 i2 x4i1 x 4 i2 x7i2 x 5 i1 x8i2 x 6 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,7),(4,8)( x5i1 x 5 i2 x6i1 x 6 i2 x9i2 x 7 i1 x10i2 x 8 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,5),(4,6)( x7i1 x 7 i2 x8i1 x 8 i2 x2i2 x 9 i1 x1i2 x 10 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,7),(4,8)( x9i1 x 9 i2 x10i1 x 10 i2 x6i2 x 2 i1 x5i2 x 1 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (1,5),(1,6)( x5i1 x 7 i2 x6i1 x 8 i2 x10i2 x 10 i1 x9i2 x 9 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,7),(4,8)( x7i1 x 9 i2 x8i1 x 10 i2 x3i2 x 4 i1 x4i2 x 3 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (1,9),(3,7), (4,8)( x91 x 1 9 x 10 1 x 2 9 x46 x 6 1 x 4 5 x 5 1 ) , ( x33 x 3 7 x 4 3 x 4 7 x107 x 8 1 x 9 7 x 7 1 ) , ( x34 x 3 8 x 4 4 x 4 8 x108 x 8 1 x 9 8 x 7 1 ) , ( x53 x 5 5 x 6 3 x 6 5 x95 x 10 1 x 10 5 x 9 1 ) ,( x54 x 5 6 x 6 4 x 6 6 x96 x 10 1 x 10 6 x 9 1 ) , ( x73 x 7 7 x 8 3 x 8 7 x27 x 3 3 x 1 7 x 4 3 ) , ( x74 x 7 8 x 8 4 x 8 8 x28 x 3 4 x 1 8 x 4 4 ) , ( x91 x 9 5 x 10 1 x 10 5 x47 x 2 1 x 3 7 x 1 1 ) ,( x91 x 9 6 x 10 1 x 10 6 x48 x 2 1 x 3 8 x 1 1 ) , ( x53 x 7 7 x 6 3 x 8 7 x37 x 10 3 x 4 7 x 9 3 ) , ( x54 x 7 8 x 6 4 x 8 8 x38 x 10 4 x 4 8 x 9 4 ) , ( x71 x 9 9 x 8 1 x 10 9 x36 x 4 1 x 3 5 x 3 1 ) ,( x73 x 9 7 x 8 3 x 10 7 x37 x 6 3 x 4 7 x 5 3 ) , ( x74 x 9 8 x 8 4 x 10 8 x38 x 6 4 x 4 8 x 5 4 ) , ( x83 x 3 5 x 10 3 x 3 6 x39 x 8 2 x 8 7 x 10 2 ) , ( x73 x 4 5 x 9 3 x 4 6 x49 x 7 2 x 7 7 x 9 2 ) ,( x83 x 3 7 x 10 3 x 3 8 x65 x 8 2 x 3 9 x 10 2 ) , ( x73 x 4 7 x 9 3 x 4 8 x55 x 7 2 x 4 9 x 9 2 ) , ( x84 x 3 5 x 10 4 x 3 6 x39 x 10 2 x 8 8 x 8 2 ) , ( x74 x 4 5 x 9 4 x 4 6 x49 x 9 2 x 7 8 x 7 2 ) ,( x84 x 3 7 x 10 4 x 3 8 x66 x 10 2 x 3 9 x 8 2 ) , ( x74 x 4 7 x 9 4 x 4 8 x56 x 9 2 x 4 9 x 7 2 ) , ( x81 x 3 9 x 10 1 x 4 9 x35 x 8 2 x 3 6 x 7 2 ) , ( x71 x 3 9 x 9 1 x 4 9 x45 x 10 2 x 4 6 x 9 2 ) . Theorem 4.4. Km ⊗Kn has an L8- decomposition if and only if mn2(m−1) ≡ 0 (mod 16). Proof. Necessity. Assume that Km ⊗Kn admits an L8- decomposition. Then the number of edges in the graph Km ⊗ Kn is mn2(m−1) 2 which should be divisible by 8, the number of edges in L8 i.e., 16|mn2(m−1) and hence mn2(m−1) ≡ 0 (mod 16). Sufficiency. We construct the required decomposition in five cases. Case(1) n ≡ 0 (mod 4). 49 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 Let n = 4s, for some s > 0. Then we can write Km ⊗Kn = ( m(m−1) 2 ) K4s,4s. Now, we get the desired decomposition by Theorem 2.7. Case(2) m ≡ 0 (mod 4), n ≡ 0 (mod 2). Subcase(i) m ≡ 0 (mod 4), n ≡ 0 (mod 4). Proof follows from Case (1). Subcase(ii) m ≡ 0 (mod 4), n ≡ 2 (mod 4). Let m = 4s and n = 6,10 for some s > 0. Now we can write Km⊗Kn = s(K4⊗Kn) ⊕ s(s−1) 2 (K4,4⊗Kn) and the graphs K4 ⊗K6,K4 ⊗K10 can be viewed as 3K6,12,3K10,20, respectively. Therefore we get the desired decomposition, by Lemma 4.1 and the Theorem 2.7. Let m = 4s and n = 4t + 2 for some s > 0, t > 2. Now we can write Km ⊗Kn = ( K4s ⊗K4(t−1) ) ⊕ ( s(s−1) 2 )( K4,4 ⊗K6 ) ⊕ 4s(4s−1)K4(t−1),6 ⊕ 3sK6,12. We get the desired decomposition, by the Case (1), Lemma 4.1 and Theorem 2.7. Case(3) m ≡ 1 (mod 4), n ≡ 0 (mod 2). Subcase(i) m ≡ 1 (mod 4), n ≡ 0 (mod 4). Proof follows from Case (1). Subcase(ii) m ≡ 1 (mod 4), n ≡ 2 (mod 4). Let m = 4s + 1 and n = 6,10 for some s > 0. Now we can write Km ⊗Kn = (K4(s−1) ⊗Kn) ⊕ K5 ⊗Kn ⊕ (s− 1)(K4,5 ⊗Kn)and the graphs K5 ⊗K6,K5 ⊗K10 can be viewed as 5K6,12,5K10,20, respectively. Therefore we get the desired decomposition, by the Case(2), Lemmas 4.2, 4.3 and Theorem 2.7. Further, let m = 4s + 1 and n = 4t + 2 for some s > 0, t > 2. Now we can write Km ⊗Kn = K4(s−1) ⊗K4t+2 ⊕ K5 ⊗ K4(t−1) ⊕ (s−1) ( K4,5 ⊗K4(t−1) ) ⊕ (s−1) ( K4,5 ⊗K6 ) ⊕ 20sK4(t−1),6 ⊕ 5K6,12. The L8- decomposition of 1st term of the above sum follows from Case (2), 2nd and 3rd term follows from Case (1) and the remaining terms of the above sum follows from the Lemma 4.2 and Theorem 2.7. Hence we get the desired decomposition. Case(4) m ≡ 0 (mod 16). Let m = 16s, for some s > 0. We can write Km ⊗Kn = s ( K16 ⊗Kn ) ⊕ ( s(s−1) 2 )( K16,16 ⊗Kn ) . The desired decomposition follows from Lemmma 4.1 and Theorems 1.3, 2.7. Case(5) m ≡ 1 (mod 16). Desired decomposition follows from Theorem 1.3 and Lemma 4.1. 5. Conclusion In this paper, we established necessary and sufficient conditions for the decomposition of ten- sor/wreath product of graphs into sunlet graphs of order 8. Further, research on the existence of such decomposition of product graphs into sunlet graphs of higher order r > 8 is under progress. Acknowledgement: The first author thank the Department of Science and Technology, Gov- ernment of India, New Delhi for its financial support through the Grant No.DST/INSPIRE Fellow- ship/2015/IF150211. The second author thank the University Grant Commission, Government of India, New Delhi for its support through the grant No.F.510/7/DRS-I/2016(SAP-I). References [1] A. D. Akwu, D. D. A. Ajayi, Decomposing certain equipartite graphs into Sunlet graphs of length 2p, AKCE Int. J. Graphs Combin. 13 (2016) 267–271. [2] B. Alspach, The wonderful Walecki construction, Bull. Inst. Combin. Appl. 52 (2008) 7–20. [3] B. Alspach, J. C. Bermond, D. Sotteau, Decomposition into cycles I: Hamilton decompositions, In: Cycles and rays (Montreal, PQ, 1987), Kluwer Academic Publishers, Dordrecht, (1990) 9–18. [4] B. Alspach, H. Gavlas, Cycle decompositions of Kn and Kn−1, J. Combin. Theory Ser. B 81(1) 50 https://doi.org/10.1016/j.akcej.2016.08.001 https://doi.org/10.1016/j.akcej.2016.08.001 https://zbmath.org/?q=an%3A1157.05035 https://doi.org/10.1007/978-94-009-0517-7_2 https://doi.org/10.1007/978-94-009-0517-7_2 https://doi.org/10.1006/jctb.2000.1996 https://doi.org/10.1006/jctb.2000.1996 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 8(1) (2021) 41–51 (2001) 77–99. [5] R. Anitha, R. S. Lekshmi, N-sun decomposition of complete graphs and complete bipartite graphs, World Acad. Sci. Eng. Tech. 27 (2007) 262-266. [6] R. Anitha, R.S. Lekshmi, N-sun decomposition of complete, complete bipartite and some Harary graphs, Int. J. Comput. Math. Sci. 2 (2008). [7] J. A. Bondy, U. R. S. Murty, Graph theory with applications, The Macmillan Press Ltd, New York (1976). [8] D. Bryant, Cycle decompositions of complete graphs, in Surveys in Combinatorics 2007, A. Hilton and J. Talbot (Editors), London Mathematical Society Lecture Note Series 346, Proceedings of the 21st British Combinatorial Conference, Cambridge University Press (2007) 67–97. [9] D. Bryant, C. A. Rodger, Cycle decompositions, C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Hand- book of Combinatorial Designs (2nd edition), CRC Press, Boca Raton (2007) 373–382. [10] D. Froncek, Decomposition of complete graphs into small graphs, Opuscula Math. 30(3) (2010) 277–280. [11] C. M. Fu, M. H. Huang, Y. L. Lin, On the existence of 5-sun systems, Discrete Math. 313(24) (2013) 2942–2950. [12] C. M. Fu, N. H. Jhuang, Y. L. Lin, H. M. Sung, From steiner triple systems to 3-sun systems, Taiwanese J. Math. 16(2) (2012) 531–543. [13] C. M. Fu, N. H. Jhuang, Y. L. Lin, H. M. Sung, On the existence of k-sun systems, Discrete Math. 312 (2012) 1931–1939. [14] M. Gionfriddo, G. Lo Faro, S. Milici, A. Tripodi, On the existence of uniformly resolvable decompo- sitions of K into 1-factors and h-suns, Utilitas Mathematica 99 (2016) 331–339. [15] A. J. W. Hilton, Hamiltonian decompositions of complete graphs, J. Combin.Theory B 36(2) (1984) 125–134. [16] A. J. W. Hilton, C. A. Rodger, Hamiltonian decompositions of complete regular s-partite graphs, Discrete Math. 58 (1986) 63–78. [17] Z. Liang, J. Guo, Decomposition of complete multigraphs into Crown graphs, J. Appl. Math. Comput. 32 (2010) 507–517. [18] Z. Liang, J. Guo, J. Wang, On the Crown graph decompositions containing odd cycle, Int. J. Comb. Graph Theory Appl. 4 (2019) 1–23. [19] C. Lin, T-W Shyu, A necessary and sufficient condition for the star decomposition of complete graphs, J. Graph Theory, 23 (1996) 361–364. [20] R. Frucht, Graceful numbering of wheels and related graphs, Ann. New York Acad. Sci 319 (1979) 219–229. [21] M. Sajna, Cycle decompositions III: Complete graphs and fixed length cycles, J. Combin. Des. 10 (2002) 27–78. [22] D. Sotteau, Decomposition of Km,n(K∗m,n) into cycles (circuits) of length 2k, J. Combin. Theory Ser. B 30(1) (1981) 75–81. [23] M. Tarsi, Decomposition of complete multigraphs into stars, Discrete Mathematics 26(3) (1979) 273–278. [24] M. Tarsi, Decomposition of complete multigraph into simple paths: Nonbalanced Handcuffed designs, J. Combin. Theory Ser. A 34 (1983) 60–70. [25] M. Truszczynski, Note on the decomposition of λKm,n(λK∗m,n) into paths, Discrete Math. 55 (1985) 89–96. [26] K. Ushio, S. Tazawa, S. Yamamoto, On claw-decomposition of complete multipartite graphs, Hi- roshima Math. J. 8(1) (1978) 207–210. [27] S. Yamamoto, H. Ikeda, S. Shige-Eda, K. Ushio, N. Hamada, On claw decomposition of complete graphs and complete bipartite graphs, Hiroshima Math. J. 5(1) (1975) 33–42. 51 https://doi.org/10.1006/jctb.2000.1996 https://doi.org/10.1006/jctb.2000.1996 https://doi.org/10.5281/zenodo.1057183 https://doi.org/10.5281/zenodo.1057183 https://doi.org/10.5281/zenodo.1058671 https://doi.org/10.5281/zenodo.1058671 http://dx.doi.org/10.7494/OpMath.2010.30.3.277 http://dx.doi.org/10.7494/OpMath.2010.30.3.277 https://doi.org/10.1016/j.disc.2013.09.007 https://doi.org/10.1016/j.disc.2013.09.007 https://www.doi.org/10.11650/twjm/1500406600 https://www.doi.org/10.11650/twjm/1500406600 https://doi.org/10.1016/j.disc.2012.03.007 https://doi.org/10.1016/j.disc.2012.03.007 https://doi.org/10.1016/0095-8956(84)90020-0 https://doi.org/10.1016/0095-8956(84)90020-0 https://doi.org/10.1016/0012-365X(86)90186-X https://doi.org/10.1016/0012-365X(86)90186-X https://doi.org/10.1007/s12190-009-0267-0 https://doi.org/10.1007/s12190-009-0267-0 https://doi.org/10.1002/(SICI)1097-0118(199612)23:4%3C361::AID-JGT5%3E3.0.CO;2-P https://doi.org/10.1002/(SICI)1097-0118(199612)23:4%3C361::AID-JGT5%3E3.0.CO;2-P https://doi.org/10.1111/j.1749-6632.1979.tb32792.x https://doi.org/10.1111/j.1749-6632.1979.tb32792.x https://doi.org/10.1002/jcd.1027 https://doi.org/10.1002/jcd.1027 https://doi.org/10.1016/0095-8956(81)90093-9 https://doi.org/10.1016/0095-8956(81)90093-9 https://doi.org/10.1016/0012-365X(79)90034-7 https://doi.org/10.1016/0012-365X(79)90034-7 https://doi.org/10.1016/0097-3165(83)90040-7 https://doi.org/10.1016/0097-3165(83)90040-7 https://doi.org/10.1016/S0012-365X(85)80023-6 https://doi.org/10.1016/S0012-365X(85)80023-6 http://doi.org/10.32917/hmj/1206135570 http://doi.org/10.32917/hmj/1206135570 http://doi.org/10.32917/hmj/1206136782 http://doi.org/10.32917/hmj/1206136782 Introduction L8- decomposition of Km,n L8- decomposition of Km Kn L8- decomposition of Km Kn Conclusion References