A Necessary and Sufficient Hall Condition for Hypergraphs

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).