Perfect graph

In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Due to the strong perfect graph theorem, perfect graphs are the same as Berge graphs. A graph G is a Berge graph if neither G nor its complement has an odd-length induced cycle of length 5 or more.

The perfect graphs include many important families of graphs, and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time. In addition, several important min-max theorems in combinatorics, such as Dilworth's theorem, can be expressed in terms of the perfection of certain associated graphs.

The theory of perfect graphs developed from a 1958 result of Tibor Gallai that in modern language can be interpreted as stating that the complement of a bipartite graph is perfect; this result can also be viewed as a simple equivalent of König's theorem, a much earlier result relating matchings and vertex covers in bipartite graphs. The first use of the phrase "perfect graph" appears to be in a 1963 paper of Claude Berge, after whom Berge graphs are named. In this paper he unified Gallai's result with several similar results by defining perfect graphs, and he conjectured the equivalence of the perfect graph and Berge graph definitions; Berge's conjecture was proved in 2002 as the strong perfect graph theorem.

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