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  "body": "\n## Statement\n\nLet $G$ be a finite [[graph|0JD5C82QMB]], and let $\\overline{G}$ be its [[complement|0RNMP6VCR2]].\nAn odd hole of a graph is an [[induced subgraph|0V05FC4GKQ]] that is a [[cycle|07CT9RDY7Q]] of odd length at least $5$.\nThen $G$ is [[perfect|1YZ7NV161A]] if and only if neither $G$ nor $\\overline{G}$ has an odd hole.\n\n## Remarks\n\nThe complement of an odd hole is called an odd antihole, and graphs with neither are called Berge graphs.\nThe easy direction holds because an odd hole $C_{2k+1}$ with $k \\ge 2$ has $\\omega = 2$ and $\\chi = 3$, and its complement has $\\omega = k$ and $\\chi = k + 1$.\nThe hard direction was conjectured by Claude Berge and proved by Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, with the proof published in 2006.\nTheir proof shows that every Berge graph lies in one of a few basic classes of perfect graphs or admits one of a few kinds of decomposition.\nA smallest Berge graph that is not perfect can have neither property.\nThe theorem implies the weak perfect graph theorem, since the condition is symmetric in $G$ and $\\overline{G}$.\n\n## Explanations\n\n### L1 human\n\nA graph is perfect when, in every part of it, the colours needed equal the size of the largest clique.\nThe theorem says there are only two kinds of obstruction: an odd cycle of length at least five with no chords, or the complement of one.\nIf you cannot find either inside a graph, the graph is perfect.\n\n### L2 human\n\nThe theorem turns a condition about all induced subgraphs into a finite list of forbidden patterns.\nIt was one of the main open problems of graph theory for about forty years.\nLater work showed that Berge graphs, and hence perfect graphs, can be recognized in polynomial time.\n",
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