发表机构
Auburn University（奥本大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究否定了Hao等人提出的‘度数为3的顶点两两距离至少3的3/2-坚韧平面三角剖分是哈密顿图’的问题，证明了存在满足任意长距离条件的此类非哈密顿图，明确哈密顿圈的障碍源于局部构型而非其邻近性。
AI 中文摘要
根据Tutte于1956年提出的经典定理，每一个4-连通平面图都是哈密顿图，且每一个阶数至少为3、坚韧度大于3/2的平面图都是哈密顿图。1999年，Owens构造了一系列极大平面图，其坚韧度从下方趋近于3/2，且这些图甚至不包含2-因子，他还提出了一个问题：是否存在坚韧度恰好为3/2且不含2-因子的极大平面图？2025年，Shan构造了一个不含2-因子的3/2-坚韧平面三角剖分，该构造中有许多对度数为3的顶点拥有共同邻点。通过对度数为3的顶点施加距离条件，Hao、Ma、Shan和Yang近期证明，每一个阶数至少为3、且度数为3的顶点两两之间距离至少为3的3/2-坚韧平面三角剖分都存在2-因子，他们还提出了一个问题：这类图是否都是哈密顿图？我们给出了该问题的否定答案，实际上证明了更强的结论：对于每一个正整数ℓ，都存在一个3/2-坚韧的非哈密顿平面三角剖分，其度数为3的顶点两两之间距离至少为ℓ。因此，尽管该距离条件保证了2-因子的存在，但并不能保证图是哈密顿图：哈密顿圈的本质障碍是涉及度数为3的顶点的某种局部构型，而非图中此类构型的 proximity（邻近性）。
英文摘要
By Tutte's classic theorem of 1956 that every 4-connected planar graph is Hamiltonian, every planar graph of order at least three with toughness greater than $\frac{3}{2}$ is Hamiltonian. In 1999, Owens constructed a sequence of maximal planar graphs whose toughness approaches $\frac{3}{2}$ from below and which do not contain even a 2-factor, and he asked whether there exists a maximal planar graph with toughness exactly $\frac{3}{2}$ and with no 2-factor. In 2025, Shan constructed a $\frac{3}{2}$-tough plane triangulation with no 2-factor. In that construction, there are many pairs of vertices of degree $3$ that have a common neighbor. By imposing a distance condition on the vertices of degree $3$, Hao, Ma, Shan, and Yang recently proved that every $\frac{3}{2}$-tough plane triangulation of order at least three whose vertices of degree $3$ are pairwise at distance at least $3$ has a 2-factor, and they asked whether every such graph is Hamiltonian. We answer this question in the negative, and in fact prove the following stronger statement: for every positive integer $\ell$, there exists a $\frac{3}{2}$-tough non-Hamiltonian plane triangulation whose vertices of degree $3$ are pairwise at distance at least $\ell$. Thus, although the distance condition guarantees the existence of a 2-factor, it does not guarantee that the graph is Hamiltonian: the essential obstruction to a Hamiltonian cycle is a certain local configuration involving a vertex of degree $3$, rather than the proximity of such configurations in the graph.
