图棱柱中的配对哈密顿性质
The Pairing-Hamiltonian property in graph prisms
AI总结：
该研究拓展了Fink关于超立方体PH性质的结论，证明具有配对哈密顿（PH）性质的图的棱柱图也具备该性质，且连通图的足够高阶次棱柱均具有PH性质。
摘要：
设$G$为偶阶图，令$K_G$为与$G$具有相同顶点集的完全图。$K_G$的一个完美匹配被称为$G$的一个配对。若对于$G$的每一个配对$M$，都能找到$G$的一个完美匹配$N$，使得$M \cup N$是$K_G$的一个哈密顿环，则称$G$具有配对哈密顿性质，简称PH性质。2007年，Fink在《J. Combin. Theory Ser. B》第97卷中证明，对任意$d\geq 2$，$d$维超立方体$\mathcal{Q}_d$具有PH性质，从而证明了Kreweras于1996年提出的一个猜想。本文拓展了Fink的结果，证明了若图$G$具有PH性质，则$G$的棱柱图$\mathcal{P}(G)$也具有PH性质。此外，我们还证明，若$G$是连通图，则存在正整数$k_0$，使得对所有$k \ge k_0$，图的$k$次棱柱$\mathcal{P}^k(G)$都具有PH性质。
英文摘要：
Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink [J. Combin. Theory Ser. B, 97] proved that for every $d\geq 2$, the $d$-dimensional hypercube $\mathcal{Q}_d$ has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph $G$ having the PH-property, the prism graph $\mathcal{P}(G)$ of $G$ has the PH-property as well. Moreover, if $G$ is a connected graph, we show that there exists a positive integer $k_0$ such that the $k^{\textrm{th}}$-prism of a graph $\mathcal{P}^k(G)$ has the PH-property for all $k \ge k_0$.
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。
