Perfect graph

Remarks

Example: every bipartite graph $G$ is perfect. An induced subgraph $H$ of $G$ is again bipartite, so $\chi(H) = \omega(H) = 2$ if $H$ has an edge, and $\chi(H) = \omega(H) = 1$ if $H$ has vertices but no edge. Non-example: the cycle $C_5$ is not perfect, because $\omega(C_5) = 2$ while $\chi(C_5) = 3$, so the condition fails for $H = C_5$ itself. The condition must hold for all induced subgraphs, not just for $G$: the disjoint union of $C_5$ and a triangle has $\chi = \omega = 3$ but is not perfect. By the weak perfect graph theorem of Lovász, the complement of a perfect graph is perfect.