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  "body": "\n## Statement\n\nLet $G$ be a finite connected simple [[graph|0JD5C82QMB]] with [[maximum degree|0KNAN9F27P]] $\\Delta$.\nSuppose that $G$ is not a [[complete graph|03WKB2H474]] and not a [[cycle graph|0QQ0F12962]] of odd length.\nThen $\\chi(G) \\le \\Delta$.\n\n## Remarks\n\nA [[greedy colouring|04AAKZNXHD]] gives $\\chi(G) \\le \\Delta + 1$ for every graph. Brooks' theorem improves this bound by $1$ in all other cases.\nThe two exceptions need $\\Delta + 1$ colours: $\\chi(K_n) = n$ with $\\Delta = n - 1$, and an odd cycle has $\\chi = 3$ with $\\Delta = 2$.\n\n## History\n\nR. Leonard Brooks published the theorem in 1941.\n\n## Explanations\n\n### L1 human\n\nIf each vertex has at most $\\Delta$ neighbours, then $\\Delta + 1$ colours are always enough.\nBrooks' theorem says that $\\Delta$ colours are enough, except for complete graphs and odd cycles.\n",
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  "name": "Brooks' theorem",
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