AI 中文总结
研究树与奇数路径笛卡尔积的哈密顿性问题，通过构造最大度为\(4\)且有\(\{P_2,P_3\}\)-因子的树\(T\)，证明对于奇数\(n\)，\(T\Box P_n\)不是哈密顿图，反驳了Kao和Weng的猜想。
AI 中文摘要
图\(G\)中的\(\{P_2,P_3\}\)-因子是指\(G\)的一个因子，其每个分量都是两个或三个顶点的路径。设\(T\Box P_n\)为树\(T\)与\(n\)个顶点路径的笛卡尔积。Kao和Weng证明，如果\(T\)有路径因子且\(n\)是足够大的偶数，则\(T\Box P_n\)是哈密顿图。本文证明，对于每个奇数\(n\)，存在一个最大度为\(4\)且有\(\{P_2,P_3\}\)-因子的树\(T\)，使得\(T\Box P_n\)不是哈密顿图，从而反驳了Kao和Weng的一个猜想。
英文摘要
A $\{P_2,P_3\}$-factor in a graph $G$ is a factor of $G$ in which every component is a path on two or three vertices. Let $T\Box P_n$ be the Cartesian product of a tree $T$ and a path on $n$ vertices. Kao and Weng proved that $T\Box P_n$ is hamiltonian if $T$ has a path factor and $n$ is a sufficiently large even integer. In this article we prove that, for every odd $n$, there exists a tree $T$ of maximum degree 4 that has a $\{P_2,P_3\}$-factor such that $T\Box P_n$ is not hamiltonian, thereby refuting a conjecture by Kao and Weng.
