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  "body": "\n## Statement\n\nLet $k \\ge 1$ be an integer, and let $G$ be a finite graph with [[chromatic number|0C5M7AM6FH]] $\\chi(G) \\ge k$.\nThen the [[complete graph|03WKB2H474]] $K_k$ is a [[minor|VYQ32ZBPEE]] of $G$.\n\n## Remarks\n\nEquivalently, every finite graph with no $K_k$ minor can be properly coloured with $k - 1$ colours.\nHugo Hadwiger stated the conjecture in 1943.\nIt is easy for $k \\le 3$, and Hadwiger proved it for $k = 4$.\nFor $k = 5$ it is equivalent to the [[four colour theorem|0M406JC8J1]], by a structure theorem of Wagner on graphs without a $K_5$ minor.\nFor $k = 6$ Robertson, Seymour and Thomas proved it in 1993, by reduction to the four colour theorem.\nThe conjecture is open for every $k \\ge 7$.\n\n## Explanations\n\n### L1 human\n\nA minor of a graph is what you get by deleting vertices and edges and by shrinking edges to points.\nHadwiger's conjecture says that a graph needs many colours only for one reason: it contains a big complete graph in this loose sense.\nThe case of five colours already contains the four colour theorem, so the conjecture is a far-reaching generalization of it.\n",
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      "first_stated": "1943",
      "posed_by": "Hugo Hadwiger"
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