The Third Proof of Lov\'asz's Cathedral Theorem

TL;DR

This paper presents a new, more natural proof of Lovász's cathedral theorem on saturated graphs with perfect matchings, avoiding the use of the Gallai-Edmonds structure theorem and revealing refined properties.

Contribution

The authors provide a novel proof of the cathedral theorem that leverages their previous work on the canonical structures of graphs with perfect matchings, simplifying and generalizing the original proof.

Findings

The new proof is more natural and avoids the Gallai-Edmonds structure theorem.

It reveals refined properties of saturated graphs with perfect matchings.

The approach generalizes the original theorem's applications.

Abstract

A graph $G$ with a perfect matching is called saturated if $G+e$ has more perfect matchings than $G$ for any edge $e$ that is not in $G$. Lov\'asz gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which revealed canonical structures of general graphs with perfect matchings. Here, the cathedral theorem is derived in quite a natural way, providing more refined or generalized properties. Moreover, the new proof shows that it can be proved without using the Gallai-Edmonds structure theorem.