The Strong Perfect Graph Conjecture for pan-free graphs

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...

About the Strong Perfect Graph Conjecture on circular partitionable graphs

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...

Combinatorial designs related to the strong perfect graph conjecture

Our graphs are “Michigan” except that they have vertices and edges rather than points and lines. If G is a graph, then y1 = y1 (G) denotes the number of its vertices, ar = (Y(G) denotes the size of its largest stable (independent) set of vertices and o = o(G) denotes the size of its largest clique. The graphs that we are interested in have the following three properties: (i) n =0X0+1, (ii) ever...

Forcing Colorations and the Strong Perfect Graph Conjecture

We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! ? 1-cliques in minimal imperfect graphs.