In graph theory, the perfect graph theorem of László Lovász (1972a, 1972b) states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by Berge (1961, 1963), and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs.
Statement A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest clique equals the minimum number of colors in a coloring of the subgraph. Perfect graphs include many important graphs classes including bipartite graphs, chordal graphs, and comparability graphs. The complement of a graph has an edge between two vertices if and only if the original graph does not have an edge between the same two vertices. Thus, a clique in the original graph becomes an independent set in the complement and a coloring of the original graph becomes a clique cover of the complement. The perfect graph theorem states:
The complement of a perfect graph is perfect. Equivalently, in a perfect graph, the size of the maximum independent set equals the minimum number of cliques in a clique cover.
Example
Let G {\displaystyle G} be a cycle graph of odd length greater than three (a so-called "odd hole"). Then G {\displaystyle G} requires at least three colors in any coloring, but has no triangle, so it is not perfect. By the perfect graph theorem, the complement of G {\displaystyle G} (an "odd antihole") must therefore also not be perfect. If G {\displaystyle G} is a cycle of five vertices, it is isomorphic to its complement, but this property is not true for longer odd cycles, and it is not as trivial to compute the clique number and chromatic number in an odd antihole as it is in an odd hole. As the strong perfect graph theorem states, the odd holes and odd antiholes turn out to be the minimal forbidden induced subgraphs for the perfect graphs.
Applications In a nontrivial bipartite graph, the optimal number of colors is (by definition) two, and (since bipartite graphs are triangle-free) the maximum clique size is also two. Also, any induced subgraph of a bipartite graph remains bipartite. Therefore, bipartite graphs are perfect. In n {\displaystyle n} -vertex bipartite graphs, a minimum clique cover takes the form of a maximum matching together with an additional clique for every unmatched vertex, with size n − M {\displaystyle n-M} , where M {\displaystyle M} is the cardinality of the matching. Thus, in this case, the perfect graph theorem implies Kőnig's theorem that the size of a maximum independent set in a bipartite graph is also n − M {\displaystyle n-M} , a result that was a major inspiration for Berge's formulation of the theory of perfect graphs. Mirsky's theorem characterizing the height of a partially ordered set in terms of partitions into antichains can be formulated as the perfection of the comparability graph of the partially ordered set, and Dilworth's theorem characterizing the width of a partially ordered set in terms of partitions into chains can be formulated as the perfection of the complements of these graphs. Thus, the perfect graph theorem can be used to prove Dilworth's theorem from the (much easier) proof of Mirsky's theorem, or vice versa.
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