In graph theory, the perfect graph theorem of states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by , 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:

1. 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

be a cycle graph of odd 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 bipartite graphs, a minimum clique cover takes the form of a maximum matching 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 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.

Lovász's proof

To prove the perfect graph theorem, Lovász used an operation of replacing vertices in a graph by cliques; it was already known to Berge that, if a graph is perfect, the graph formed by this replacement process is also perfect. has clique number and chromatic number one less than that of the given graph. The removed vertices and the new copy of the doubled vertex can then be added back as a single color class, showing that in this case the doubling step leaves the chromatic number unchanged. The same argument shows that doubling preserves the equality of the clique number and the chromatic number in every induced subgraph of the given graph, so each doubling step preserves the perfection of the graph.

are all disjoint, and the result follows by induction on this number.

Relation to the strong perfect graph theorem

The strong perfect graph theorem of states that a graph is perfect if and only if none of its induced subgraphs are cycles of odd length greater than or equal to five, or their complements. Because this characterization is unaffected by graph complementation, it immediately implies the weak perfect graph theorem.

Generalizations

proved that, if the edges of a complete graph are partitioned into three subgraphs in such a way that every three vertices induce a connected graph in one of the three subgraphs, and if two of the subgraphs are perfect, then the third subgraph is also perfect. The perfect graph theorem is the special case of this result when one of the three subgraphs is the empty graph.

Notes

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