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  "body": "\n## Statement\n\nLet $G$ be a finite simple [[graph|0JD5C82QMB]] with $n \\ge 3$ vertices.\nSuppose that every vertex of $G$ has [[degree|0T0F6CCS9W]] at least $n/2$.\nThen $G$ is [[Hamiltonian|090WPKFABX]].\n\n## Remarks\n\nThe hypothesis says that the [[minimum degree|0TKZCSGCCA]] is at least $n/2$.\nThe theorem follows from [[Ore's theorem|0NACHJYPKW]]: if all degrees are at least $n/2$, then each sum of two degrees is at least $n$.\nThe bound is sharp. The complete bipartite graph $K_{k, k+1}$ has minimum degree $k \u003c (2k+1)/2$, and it is not Hamiltonian.\n\n## History\n\nG. A. Dirac published the theorem in 1952.\n\n## Explanations\n\n### L1 human\n\nIf each vertex is joined to at least half of the other vertices, there is a round trip through all vertices.\nA graph with this many edges cannot avoid a Hamiltonian cycle.\n",
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  "name": "Dirac's theorem on Hamiltonian cycles",
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