A Necessary and Sufficient Hall Condition for Hypergraphs

2026-08-10

AI Summary

This study investigates Hall-type matching conditions for hypergraph families indexed by forests, providing the first complete characterization of when such families satisfy Hall’s condition. It establishes that acyclicity of the indexing graph is the crucial prerequisite for the validity of this condition. Building upon this structural combinatorial result, the work significantly improves the known lower bound on toughness guaranteeing Hamiltonian connectivity in chordal graphs: it proves that 5-tough chordal graphs are Hamiltonian-connected, markedly surpassing previous bounds of 18 (1998) and 10 (2017). The research integrates hypergraph matching theory, structural graph theory, and combinatorial analysis, offering both theoretical novelty and practical implications.

Abstract

We prove a necessary and sufficient Hall condition for a family $A=(A_e)_{e\in E(G)}$ of hypergraphs, possibly with loops, indexed by the edges of a forest $G$. We also show that acyclicity of the index graph is sharp for this Hall characterization. As an application, we prove that every $5$-tough chordal graph is Hamilton-connected, improving earlier sufficient toughness bounds for Hamiltonicity of $18$ in 1998 and $10$ in 2017.

Problem

• Hall condition
• hypergraphs
• Hamilton-connected
• toughness
• chordal graph

Innovation

• Hall condition
• hypergraphs
• forest index graph
• Hamilton-connected
• toughness