The strong perfect graph conjecture

TL;DR

This paper discusses the proof of the strong perfect graph conjecture, which characterizes perfect graphs by the absence of odd holes and their complements, a significant milestone in graph theory.

Contribution

It provides an overview and insight into the recent proof of the long-standing strong perfect graph conjecture by Chudnovsky et al.

Findings

Proof of the strong perfect graph conjecture completed in 2002

Characterization of perfect graphs via forbidden induced subgraphs

Advancement in understanding graph perfection and structure

Abstract

A graph is {\em perfect} if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called {\em basic}. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by Lov\'asz. This result is now known as the perfect graph theorem. The strong perfect graph conjecture (SPGC) states that a graph is perfect if and only if it does not contain an odd hole or its complement. The SPGC has attracted a lot of attention. It was proved recently (May 2002) in a remarkable sequence of results by Chudnovsky, Robertson, Seymour and Thomas. The proof is difficult and, as of this writing, they are still checking the details. Here we give a flavor of the proof.