Odd Pairs of Cliques

Abstract

A graph is Berge if it has no induced odd cycle on at least 5 vertices and no complement of induced odd cycle on at least 5 vertices. A graph is perfect if the chromatic number equals the maximum clique number for every induced subgraph. Chudnovsky, Robertson, Seymour and Thomas proved that every Berge graph either falls into some classical family of perfect graphs, or has a structural fault that cannot occur in a minimal imperfect graph. A corollary of this is the strong perfect graph theorem conjectured by Berge: every Berge graph is perfect. An even pair of vertices in a graph is a pair of vertices such that every induced path between them has even length. Meyniel proved that a minimal imperfect graph cannot contain an even pair. So even pairs may be considered as a structural fault. Chudnovsky et al. do not use them, and it is known that some classes of Berge graph have no even pairs. The aim of this work is to investigate an "even-pair-like" notion that could be a structural fault present in every Berge graph. An odd pair of cliques is a pair of cliques $\{K_1, K_2\}$ such that every induced path from $K_1$ to $K_2$ with no interior vertex in $K_1 \cup K_2$ has odd length. We conjecture that for every Berge graph $G$ on at least two vertices, either one of $G, \bar{G}$ has an even pair, or one of $G, \bar{G}$ has an odd pair of cliques. We conjecture that a minimal imperfect graph has no odd pair of maximal cliques. We prove these conjectures in some special cases. We show that adding all edges between any 2 vertices of the cliques of an odd pair of cliques is an operation that preserves perfectness.