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7 - Perfect graphs

Published online by Cambridge University Press:  05 May 2015

By
Nicolas Trotignon
Edited by
Lowell W. Beineke  and
Robin J. Wilson
Nicolas Trotignon
Affiliation:
University Paris Diderot
Lowell W. Beineke
Affiliation:
Purdue University, Indiana
Robin J. Wilson
Affiliation:
The Open University, Milton Keynes
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Summary

Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization. Berge made a conjecture about them (now called the strong perfect graph theorem or SPGT) which was proved by Chudnovsky, Robertson, Seymour and Thomas in 2002. This survey about perfect graphs mostly focuses on the SPGT.

Introduction

Every graph G clearly satisfies χ(G) ≥ ω(G), where ω(G) is the clique number of G, because the vertices of a clique must receive different colours. A graph G is perfect if every induced subgraph H of G satisfies χ(H) = ω(H). A chordless cycle of length 2k + 1, for k ≥ 2, satisfies 3 = χ > ω = 2, and its complement satisfies k + 1 = χ > ω = k; these graphs are therefore imperfect. Since perfect graphs are closed under taking induced subgraphs, they must be defined by excluding a family F of graphs as induced subgraphs. The strong perfect graph theorem (SPGT for short) states that the two examples just given are the only members of F.

Let us make this more formal. A hole in a graph G is an induced cycle of length at least 4, and an antihole is a hole of G. A graph is Berge if it does not contain an odd hole or an odd antihole. The following was conjectured by Berge [3] in the 1960s and was the object of much research until it was finally proved in 2002 by Chudnovsky, Robertson, Seymour and Thomas [13].

Theorem 1.1 (Strong perfect graph theorem) A graph is perfect if and only if it is Berge.

One direction of the proof is easy: every perfect graph is Berge since, as we observed above, odd holes and antiholes satisfy χ = ω + 1. The proof of the converse statement is very long and relies on structural graph theory.

Type
Chapter
Information
Topics in Chromatic Graph Theory , pp. 137 - 160
Publisher: Cambridge University Press
Print publication year: 2015

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