Split-Neighborhood Graphs and the Strong Perfect Graph Conjecture

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

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.

Even pairs and the strong perfect graph conjecture

We will characterize all graphs that have the property that the graph and its complement are minimal even pair free. This characterization allows a new formulation of the Strong Perfect Graph Conjecture. The reader is assumed to be familiar with perfect graphs (see for example [2]). A hole is a cycle of length at least five. An odd hole is a hole that has an odd number of vertices. An (odd) ant...

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...