About Skew Partitions in Minimal Imperfect Graphs

For terms not defined here, the reader is referred to [6]. For a graph G=(V, E), we denote by G the complement of G. We use |(G) to denote the size of a largest clique in G and :(G) to denote the size of a largest stable set in G, or simply | and : when no confusion is possible. A graph G is said to be perfect if, for each induced subgraph H of G, H can be coloured with |(H) colours such that every two adjacent vertices have different colours. A graph is minimal imperfect if it is not perfect but all of its proper induced subgraphs are perfect. One can easily check that an odd chordless cycle with at least five vertices (usually called an odd hole) and its complement (usually called an odd antihole) are minimal imperfect graphs. Berge [1] conjectured that the only minimal imperfect graphs are the odd holes and the odd antiholes. This conjecture is still unsettled and known as the strong perfect graph conjecture (SPGC). Berge also conjectured that a graph is perfect if and only if its complement is perfect. This conjecture is a consequence of the following theorem of Lova sz [9]: