$C_4$-decomposition of the tensor product of complete graphs

Abstract

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 3 ⊕ ... ⊕ H k , where H 1 , H 2 , H 3 , ..., H k are all isomorphic to H , then G is said to be H -decomposable. Futhermore, 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 the necessary and sufficient conditions for the existence of C 4 -decomposition of K m × K n . Using these conditions it can be shown that every even regular complete multipartite graph G is C 4 - decomposable if the number of edges of G is divisible by 4.