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  "body": "\n## Summary\n\nA perfect graph is a graph in which, for every induced subgraph, the [[chromatic number|0VHJRXNAAS]] equals the size of the largest [[clique|0BYX1D7C2E]].\nFor every graph the chromatic number is at least the [[clique number|04MQSQ1F5Q]], and perfect graphs are those where this lower bound is always exact.\nBipartite graphs, interval graphs and comparability graphs are perfect.\nThe [[strong perfect graph theorem|MFNNZQ484T]] characterizes perfect graphs by forbidden induced subgraphs.\nOn perfect graphs, colouring and finding a largest clique can be done in polynomial time.\n\n## Explanations\n\n### L1 human\n\nTo colour a graph you need at least as many colours as the largest group of mutually adjacent vertices.\nIn a perfect graph this is always enough, not just for the whole graph but for every piece of it you get by deleting vertices.\nThe $5$-cycle is the smallest graph that is not perfect: it has no triangle, so its largest clique has size $2$, but it needs $3$ colours.\n",
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