Ear-Slicing for Matchings in Hypergraphs

TL;DR

This paper investigates matchings in factor-critical graphs and applies the findings to partition 3-regular, 3-uniform hypergraphs into triangles and edges, offering new insights into hypergraph packings and related theorems.

Contribution

It introduces a novel approach called ear-slicing for analyzing matchings in hypergraphs and extends classical theorems to hypergraph variants.

Findings

Partitioning of hypergraphs into triangles and edges achieved

New proof for a sharpening of Lu's theorem on antifactors

Hypergraph variant of Petersen's theorem established

Abstract

We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a $3$-regular, $3$-uniform hypergraph into at most one triangle (hyperedge of size $3$) and edges (subsets of size $2$ of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of Cornu\'ejols' and Pulleyblank's. The existence of such a packing can be considered to be a hypergraph variant of Petersen's theorem on perfect matchings, and leads to a simple proof for a sharpening of Lu's theorem on antifactors of graphs.