发表机构
Louisiana State University（路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造非哈密顿弦生成超图，结合弦图的哈密顿性定理，证明10-韧性图为分数哈密顿图，进而证实Devriendt关于10-韧性图为电阻正图的猜想，同时构造出韧性接近3/2的非电阻非负图。
AI 中文摘要
一个图是分数哈密顿图，当且仅当它存在一个取值在[0,1]内的非负边权重分配，使得总权重等于其顶点数，且每个非平凡边割的权重至少为2。受Chvátal韧性猜想的启发，Scheinerman和Ullman提出猜想：每个2-韧性图都是分数哈密顿图。本文证明，每个至少含3个顶点且非分数哈密顿的连通图，都存在一个非哈密顿的弦生成超图。由于添加边不会降低韧性，结合Kabela与Kaiser的定理（每个至少含3个顶点的10-韧性弦图是哈密顿图），可推出每个至少含3个顶点的10-韧性图都是分数哈密顿图。将该结果应用于电阻曲率，本文证明每个分数哈密顿图都是电阻正（RP）图，因此每个10-韧性图都是RP图，从而证实了Devriendt的猜想。另一方面，对任意ε>0，本文构造了一个非电阻非负且韧性大于3/2-ε的图，扩展了Agrahari、Bibby、Boros、Garcia、Heidercheidt与Wang近期的构造。
英文摘要
A graph is fractionally Hamiltonian if it admits a nonnegative edge weighting in $[0,1]$ of total weight equal to its order such that every nontrivial edge cut has weight at least two. Motivated by Chvátal's Toughness Conjecture, Scheinerman and Ullman conjectured that every $2$-tough graph is fractionally Hamiltonian. In this paper, we show that every connected graph on at least three vertices that is not fractionally Hamiltonian has a non-Hamiltonian chordal spanning supergraph. Since adding edges does not decrease toughness, a theorem of Kabela and Kaiser that every $10$-tough chordal graph on at least three vertices is Hamiltonian yields that every $10$-tough graph on at least three vertices is fractionally Hamiltonian. We apply this result to resistance curvature. We prove that every fractionally Hamiltonian graph is resistance positive (RP), and consequently every $10$-tough graph is RP, confirming a conjecture of Devriendt. In the other direction, for every $\varepsilon>0$, we construct a graph that is not resistance nonnegative and has toughness greater than $3/2-\varepsilon$, extending a recent construction of Agrahari, Bibby, Boros, Garcia, Heidercheidt, and Wang.
Comments16 pages, 2 figures
