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 (clique number). Equivalently stated in symbolic terms an arbitrary graph G {displaystyle G} is perfect if and only if we have: ∀ S ⊆ V ( G ) ( χ ( G [ S ] ) = ω ( G [ S ] ) ) . {displaystyle forall Ssubseteq V(G)left(chi (G)=omega (G) ight).} 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 (clique number). Equivalently stated in symbolic terms an arbitrary graph G {displaystyle G} is perfect if and only if we have: ∀ S ⊆ V ( G ) ( χ ( G [ S ] ) = ω ( G [ S ] ) ) . {displaystyle forall Ssubseteq V(G)left(chi (G)=omega (G) ight).} 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.