  Let $G$ be a simple and finite graph. A graph is said to be
\textit{decomposed} into subgraphs $H_1$ and $H_2$ which is denoted by $G= H_1
\oplus H_2$, if $G$ is the edge disjoint union of $H_1$ and $H_2$. If $G= H_1
\oplus H_2 \oplus H_3 \oplus \cdots \oplus H_k$, 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\times 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
(respectively, $C_6$-decomposition ) of $K_m \times K_n$. Using these
conditions it can be shown that every even regular complete multipartite graph
$G$ is $C_4$-decomposable (respectively, $C_6$-decomposable) if the number of
edges of $G$ is divisible by $4$ (respectively, $6$).

Combinatorics