Back Perfekter Graph German Grafo perfecto Spanish گراف تام Persian Graphe parfait French גרף מושלם HE Perfekt gráf Hungarian Grafo perfetto Italian パーフェクトグラフ Japanese 완벽 그래프 Korean Perfecte graaf Dutch
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
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, despite their greater complexity for non-perfect graphs. In addition, several important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem on matchings, and the Erdős–Szekeres theorem on monotonic sequences, can be expressed in terms of the perfection of certain associated graphs.
The perfect graph theorem states that the complement graph of a perfect graph is also perfect. The strong perfect graph theorem characterizes the perfect graphs in terms of certain forbidden induced subgraphs, leading to a polynomial time algorithm for testing whether a graph is perfect.