C6-decompositions of the tensor product of complete graphs

Let G be a simple and finite graph. A graph is said to be decomposed into subgraphs H 1 and H 2 which is denoted by G = H 1 ⊕ H 2 , if G is the edge disjoint union of H 1 and H 2 . If G = H 1 ⊕ H 2 ⊕ ⋯ ⊕ H k , where H 1 , H 2 , …, H k are all isomorphic to H , then G is said to be H -decomposable. Furthermore, if H is a cycle of length m then we say that G is C m -decomposable and this can be written as C m | G . Where G × H denotes the tensor product of graphs G and H , in this paper, we prove that the necessary conditions for the existence of C 6 -decomposition of K m × K n are sufficient. Using these conditions it can be shown that every even regular complete multipartite graph G is C 6 -decomposable if the number of edges of G is divisible by 6 .