A Graph Theoretic Proof of the Tight Cut Lemma

TL;DR

This paper offers a new purely graph theoretic proof of the Tight Cut Lemma, a fundamental result in matching theory, using canonical decomposition and towers to analyze bricks, avoiding linear programming or broader concepts.

Contribution

It introduces a novel, purely graph theoretic proof of the Tight Cut Lemma, utilizing canonical decomposition and towers, simplifying the understanding of bricks in matching theory.

Findings

Provides a purely graph theoretic proof of the Tight Cut Lemma

Develops the concept of towers for analyzing bricks

Simplifies the proof by avoiding linear programming methods

Abstract

In deriving their characterization of the perfect matchings polytope, Edmonds, Lov\'asz, and Pulleyblank introduced the so-called {\em Tight Cut Lemma} as the most challenging aspect of their work. The Tight Cut Lemma in fact claims {\em bricks} as the fundamental building blocks that constitute a graph in studying the matching polytope and can be referred to as a key result in this field. Even though the Tight Cut Lemma is a matching \textup{(}$1$-matching\textup{)} theoretic statement that consists of purely graph theoretic concepts, the known proofs either employ a linear programming argument or are established upon results regarding a substantially wider notion than matchings. This paper presents a new proof of the Tight Cut Lemma, which attains both of the two reasonable features for the first time, namely, being {\em purely graph theoretic} as well as {\em purely matching theory closed}. Our proof uses, as the only preliminary result, the canonical decomposition recently introduced by Kita. By further developing this canonical decomposition, we acquire a new device of {\em towers} to analyze the structure of bricks, and thus prove the Tight Cut Lemma. We believe that our new proof of the Tight Cut Lemma provides a highly versatile example of how to handle bricks.