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  "body": "\n## Proof\n\nWe deduce the theorem from Ore's theorem.\n\n1. Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n \\ge 3$ vertices, and suppose that $\\deg(w) \\ge n/2$ for every vertex $w$, where $\\deg$ is the [[degree|0T0F6CCS9W]].\n2. Let $u$ and $v$ be two distinct vertices of $G$ that are not adjacent.\n3. By Step 1, $\\deg(u) + \\deg(v) \\ge n/2 + n/2 = n$.\n4. So $G$ satisfies the hypothesis of [[Ore's theorem|0NACHJYPKW]]: it has $n \\ge 3$ vertices, and every pair of distinct nonadjacent vertices has degree sum at least $n$.\n5. By Ore's theorem, $G$ is [[Hamiltonian|090WPKFABX]].\n\n## Idea\n\nDirac's condition bounds each degree from below by $n/2$. Adding two such bounds gives Ore's condition on pairs of nonadjacent vertices. So Dirac's theorem is the special case of Ore's theorem where every vertex is \"rich\" on its own.\n",
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