The Strong Perfect Graph Conjecture

A graph is 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 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 Lovász. 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. Let us call Berge graph a graph that does not contain an odd hole or its complement. Conforti, Cornuéjols, Robertson, Seymour, Thomas and Vušković (2001) conjectured a structural property of Berge graphs that implies the SPGC: Every Berge graph G is basic or has a skew partition or a homogeneous pair, or G or its complement has a 2-join. A skew partition is a partition of the vertices into nonempty sets A,B, C, D such that every vertex of A is adjacent to every vertex of B and there is no edge between C and D. Chvátal introduced this concept in 1985 and conjectured that no minimally imperfect graph has a skew partition. This conjecture was proved recently by Chudnovsky and Seymour (May 2002). Cornuéjols and Cunningham introduced 2-joins in 1985 and showed that they cannot occur in a minimally imperfect graph different from an odd hole. Homogeneous pairs were introduced in 1987 by Chvátal and Sbihi, who proved that they cannot occur in minimally imperfect graphs. Since skew partitions, 2-joins and homogeneous pairs cannot occur in minimally imperfect Berge graphs, the structural property of Berge graphs stated above implies the SPGC. This structural property was proved: (i) When G contains the line graph of a bipartite subdivision of a 3connected graph (Chudnovsky, Robertson, Seymour and Thomas (September 2001)); (ii) When G contains a stretcher (Chudnovsky and Seymour (January This work was supported in part by NSF grant DMI-0098427 and ONR grant N00014-97-10196. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA 15213, USA. E-mail: