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  "body": "\n## Statement\n\nThe [[complete graph|03WKB2H474]] $K_5$ and the [[complete bipartite graph|0V0GC7BQAN]] $K_{3,3}$ are not [[planar|029JQEK15B]].\n\n## Remarks\n\nThe graph $K_5$ has $5$ vertices and $10$ edges. The [[edge bound for planar graphs|0SM1REAVYJ]] allows at most $3 \\cdot 5 - 6 = 9$ edges.\nThe graph $K_{3,3}$ has $6$ vertices and $9$ edges, and it has no cycle of length $3$. The bound $e \\le 2v - 4$ allows at most $8$ edges.\nThe statement for $K_{3,3}$ is the three utilities problem: three houses cannot each be joined to three utilities without crossings.\n\n## Explanations\n\n### L1 human\n\nTry to join each of three houses to each of three utilities with lines that never cross.\nIt is impossible, and the reason is a count of edges and faces.\n",
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