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  "body": "\n## Statement\n\nLet $G$ be a finite [[bipartite graph|07DVHJ8N0A]].\nThen the largest number of edges in a [[matching|0Z87BHAFTB]] of $G$ is equal to the smallest number of vertices in a [[vertex cover|0KDADE9RQ8]] of $G$.\n\n## Remarks\n\nIn every graph, a vertex cover has at least as many vertices as a matching has edges. Kőnig's theorem says that equality holds for bipartite graphs.\nThe theorem fails for other graphs. The triangle $K_3$ has matchings with $1$ edge only, but each vertex cover has $2$ vertices.\n\n## History\n\nDénes Kőnig published the theorem in 1931.\nJenő Egerváry found it independently in the same year, in a more general form with weights.\n\n## Explanations\n\n### L1 human\n\nIn a bipartite graph, the most pairs that you can match at once is the same as the fewest vertices that touch every edge.\nEach of the two numbers gives a certificate that the other one is optimal.\n",
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