AI 中文总结
该研究证明了带环超图族的充要霍尔条件，将其应用后得出5-坚韧弦图为哈密顿连通的结论，改进了此前哈密顿性的充分坚韧度界。
AI 中文摘要
我们证明了由森林G的边索引的超图族A=(Aₑ)_{e∈E(G)}（可能带环）的充要霍尔条件，还表明该霍尔刻画中索引图的无环性是严格的。作为应用，我们证明每个5-坚韧的弦图都是哈密顿连通的，改进了1998年哈密顿性充分坚韧度界18及2017年的10。
英文摘要
We prove a necessary and sufficient Hall condition for a family $A=(A_e)_{e\in E(G)}$ of hypergraphs indexed by the edges of a forest \(G\). This restriction on the index graph is sharp. The loop-only case recovers the classical Hall's Theorem with multiplicities for arbitrary finite set systems, while the loopless case shows that full rainbow matching is polynomial time solvable under this forest structure, although the problem is NP-complete in general. As another application, we prove every $5$-tough chordal graph is Hamilton-connected, improving toughness bounds of $18$ for Hamiltonicity (1998) and $10$ for Hamilton-connectedness (2017).
Comments17 pages, 3 figures; new content in sections 1 and 4
