Perfect graphs

Definition. $G$ is perfect if $\chi(H)=\omega(H)$ for every induced subgraph $H$.

Strong Perfect Graph Theorem (Chudnovsky–Robertson–Seymour–Thomas 2006). $G$ is perfect iff neither $G$ nor $\overline{G}$ contains an induced odd cycle of length $\ge5$.

Examples. Bipartite graphs are perfect; $C_5$ is not.

Conclusion. The SPGT settled Berge's conjecture and is a landmark in structural graph theory.