AI 中文总结
研究正则均匀可迹非哈密顿图，探讨其阶数实现问题。通过研究，证明对于任意$k\geq6$和$n\geq6(k - 2)$，存在相应阶数的$k$正则均匀可迹非哈密顿图，完善了此类图的相关研究。
AI 中文摘要
如果一个图的每个顶点都是哈密顿路径的端点，那么这个图就是均匀可迹的。Chartrand、Gould和Kapoor（1979）证明了对于每个$n\geq9$的阶数，存在不规则的均匀可迹非哈密顿图。Hu和Zhan（2022年，《离散应用数学》）考虑了$3$正则和$4$正则的情况，并询问$k$正则均匀可迹非哈密顿图可以实现哪些阶数$n$。最近，Liu和Qiao（2026年，《离散应用数学》）表明，如果$k\geq5$是奇数且$q\in\{0,2,4,6\}$，或者$k\geq6$是偶数且$q\in\{0,1,\cdots,6\}$，那么$n=p(k - 1)+q\geq6(k - 1)+q$是可以实现的。在本文中，我们表明对于任何$k\geq6$和$n\geq6(k - 2)$，存在一个阶数为$n$的$k$正则均匀可迹非哈密顿图。
英文摘要
A graph is homogeneously traceable if each vertex is an endpoint of a Hamiltonian path. Chartrand, Gould, and Kapoor (1979) proved irregular homogeneously traceable nonhamiltonian graphs exist for every order $n\ge 9$. Hu and Zhan (DAM, 2022) considered the $3$-regular and $4$-regular cases and asked which order $n$ can be realized by a $k$-regular homogeneously traceable nonhamiltonian graph. Recently, Liu and Qiao (DAM, 2026) showed that $n=p(k-1)+q\ge 6(k-1)+q$ can be realized if $k\ge 5$ is odd and $q\in\{0,2,4,6\}$, or $k\ge 6$ is even and $q\in\{0,1,...,6\}$. In this paper, we show that for any $k\ge 6$ and $n\ge 6(k-2)$, there exists a $k$-regular homogeneously traceable nonhamiltonian graph of order $n$.
