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.
In any graph, the clique number provides a lower bound for the chromatic number, as all vertices in a clique must be assigned distinct colors in any proper coloring. The perfect graphs are those for which this lower bound is tight, not just in the graph itself but in all of its induced subgraphs. For more general graphs, the chromatic number and clique number can differ; e.g., a cycle of length 5 requires three colors in any proper coloring but its largest clique has size 2.
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 (Grötschel, Lovász & Schrijver 1988). 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. In this paper he unified Gallai's result with several similar results by defining perfect graphs and conjecturing a characterization of these graphs that was later proven as the Strong Perfect Graph Theorem.
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Some of the more well-known perfect graphs are
Characterization of perfect graphs was a longstanding open problem. The first breakthrough was the affirmative answer to the then perfect graph conjecture.
Perfect Graph Theorem. (Lovász 1972)
An induced cycle of odd length at least 5 is called an odd hole. An induced subgraph that is the complement of an odd hole is called an odd antihole. A graph that does not contain any odd holes or odd antiholes is called a Berge graph. By virtue of the perfect graph theorem, a perfect graph is necessarily a Berge graph. But it puzzled people for a long time whether the converse was true. This was known as the strong perfect graph conjecture and was finally answered in the affirmative in May, 2002.
Strong Perfect Graph Theorem. (Chudnovsky, Robertson, Seymour, Thomas 2002)
As with many results discovered through structural methods, the theorem's proof is long and technical. Efforts towards solving the problem have led to deep insights in the field of structural graph theory, where many related problems remain open. For example, it was proved some time ago that the problem of recognizing Berge graphs is in co-NP (Lovász 1983), but it was not known whether or not it is in P until after the proof of the Strong Perfect Graph Theorem appeared. (A polynomial time algorithm was discovered by Chudnovsky, Cornuéjols, Liu, Seymour, and Vušković shortly afterwards, independent of the Strong Perfect Graph Theorem.)