Partitionable graphs, circle graphs, and the berge strong perfect graph conjecture

About the Strong Perfect Graph Conjecture on circular partitionable graphs

Perfect Graphs, Partitionable Graphs and Cutsets

A graph G is perfect if, for all induced subgraphs of G, the size of a largest clique is equal to the chromatic number. A graph is minimally imperfect if it is not perfect but all its proper induced subgraphs are. A hole is a chordless cycle of length at least four. The strong perfect graph conjecture of Berge [1] states that G is minimally imperfect if and only if G or its complement is an odd...

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...

Chair-Free Berge Graphs Are Perfect

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd, eb}. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect...

The strong perfect graph conjecture holds for diamonded odd cycle-free graphs

We define a diamonded odd cycle to be an odd cycle C with exactly two chords and either a) C has length five and the two chords are non-crossing; or b) C has length greater than five and has chords (x,y) and (x,z) with (y,z) an edge of C and there exists a node w not on C adjacent to y and C, but not x. In this paper, we show that given a diamonded odd cycle-free graph G, G is perfect if and on...