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  "body": "\n## Definition\n\nLet $G$ be a [[graph|0JD5C82QMB]].\nA perfect matching of $G$ is a [[matching|0Z87BHAFTB]] $M$ such that every vertex of $G$ is an end of an edge of $M$.\n\n## Remarks\n\nIn a perfect matching, each vertex is an end of exactly one edge of $M$. Thus a finite graph with a perfect matching has an even number of vertices.\n[[Hall's theorem for bipartite graphs|0X2GJPNPFN]] characterizes the bipartite graphs with a perfect matching. [[Tutte's theorem|08J7R35B6Q]] does this for all finite graphs.\nThe [[complete graph|03WKB2H474]] $K_4$ has $3$ perfect matchings.\n\n## Explanations\n\n### L1 human\n\nA perfect matching pairs up all vertices of a graph along edges, with nobody left alone.\nFor this, the number of vertices must be even.\n",
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