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  "body": "\n## Definition\n\nLet $G$ be a finite [[graph|0JD5C82QMB]].\nFor a graph $H$, let $\\chi(H)$ be its [[chromatic number|0C5M7AM6FH]] and $\\omega(H)$ its [[clique number|0WCKRBHFFD]].\nThe graph $G$ is perfect if $\\chi(H) = \\omega(H)$ for every [[induced subgraph|0V05FC4GKQ]] $H$ of $G$.\n\n## Remarks\n\nExample: every bipartite graph $G$ is perfect.\nAn 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.\nNon-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.\nThe 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.\nBy the weak perfect graph theorem of Lovász, the complement of a perfect graph is perfect.\n",
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