完全图张量积的$C_4$和$C_6$分解
$C_4$ and $C_6$ decomposition of the tensor product of complete graphs
摘要：
设$G$是一个简单的有限图。一个图称为\textit{decomposed}，它分为子图$H_1$和$H_2$，用$G= H_1 \oplus H_2$表示，如果$G$是$H_1$和$H_2$的边不交并.如果是$G= H_1 \oplus H_2 \oplus H_3 \oplus \cdots \oplus H_k$，其中\ $H_1$，$H_2$，$H_3$，.，$H_k$都同构于$H$，则称$G$是$H$ -可分解的。因此，如果$H$是一个长度为$m$的循环，那么我们说$G$是$C_m$可分解的，并且可以写为$C_m|G$。这里$ G\times H$表示图$G$和$H$的张量积，本文证明了$K_m \times K_n$的$C_4$ -分解（分别为$C_6$ -分解）存在的充要条件.利用这些条件，可以证明，如果$G$的边数可被$4$（分别为$6$）整除，则每个偶正则完全多部图$G$都是$C_4$ -可分解的（分别为$C_6$ -可分解的）。
英文摘要：
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, $6$).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
