On minimal imperfect NP5 graphs

We prove that a minimal imperfect graph containing a vertex which is not on any P 5 has no odd pair. A graph is perfect if the vertices of any induced subgraph H can be colored, in such a way that no two adjacent vertices receive the same color, with a number of colors (denoted by (H)) not exceeding the cardinality !(H) of a maximum clique of H. A graph is minimal imperfect if all its proper induced subgraphs are perfect but it is not. It is an easy task to check that an odd chordless cycle of length at least ve (usually called a hole), as well as its complement (usually called an anti-hole) are minimal imperfect graphs. This remark and some early results concerning perfect graphs determined Berge 1] to formulate the two following conjectures (known as the Strong and the Weak Perfect Graph Conjecture) (SPGC) A graph is perfect if and only if it does not contain an odd hole or an odd anti-hole as an induced subgraph.