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  "body": "\n## Statement\n\nLet $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $V$.\nFor $U \\subseteq V$, let $o(G - U)$ be the number of [[connected components|01VVP10MWJ]] of $G - U$ with an odd number of vertices.\nThen $G$ has a [[perfect matching|05ZPH8KF9K]] if and only if $o(G - U) \\le |U|$ for every $U \\subseteq V$.\n\n## Remarks\n\nThe condition is necessary. In a perfect matching, each odd component of $G - U$ has a vertex that is matched to a vertex of $U$. These vertices of $U$ are distinct.\nFor $U = \\varnothing$, the condition says that each component of $G$ has an even number of vertices.\nThe theorem is a special case of the Tutte–Berge formula for the size of a maximum matching.\n\n## History\n\nThe theorem is named after William Thomas Tutte.\n\n## Explanations\n\n### L1 human\n\nDelete some vertices, and count the pieces with an odd number of vertices that remain.\nEach odd piece needs a partner among the deleted vertices, and Tutte's theorem says this is the only obstruction.\n",
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