Some conjectures on perfect graphs

The complement of a graph G is denoted by G. χ(G) denotes the chromatic number and ω(G) the clique number of G. The cycles of odd length at least five are called odd holes and the complements of odd holes are called odd anti-holes. A graph G is called perfect if, for each induced subgraph G of G, χ(G) = ω(G). Classical examples of perfect graphs consist of bipartite graphs, chordal graphs and comparability graphs. Examples of nonperfect graphs are the odd holes and odd anti-holes. The most important result on perfect graphs is the following one, due to L. Lovász. The Perfect Graph Theorem used to be the so-called weak perfect graph conjecture posed by C. Berge around 1960s. His stronger conjecture on perfect graphs, which is still open, is as follows. The Strong Perfect Graph Conjecture. Graphs without induced odd holes and odd anti-holes are perfect.