On the 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...

On the oriented perfect path double cover conjecture

‎An  oriented perfect path double cover (OPPDC) of a‎ ‎graph $G$ is a collection of directed paths in the symmetric‎ ‎orientation $G_s$ of‎ ‎$G$ such that‎ ‎each arc‎ ‎of $G_s$ lies in exactly one of the paths and each‎ ‎vertex of $G$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete‎ ‎Math‎. ‎276 (2004) 287-294) conjectured that ...

About the Strong Perfect Graph Conjecture on circular partitionable graphs

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.