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  "body": "\n## Statement\n\nLet $G$ be a finite simple connected [[planar graph|029JQEK15B]] with $v \\ge 3$ vertices and $e$ edges.\nThen $e \\le 3v - 6$.\n\n## Remarks\n\nProof idea: each face is bounded by at least $3$ edges, and each edge touches at most $2$ faces. Thus $3f \\le 2e$, and [[Euler's formula|0W1C598HMS]] gives the bound.\nIf $G$ also has no cycle of length $3$, then $e \\le 2v - 4$.\nA consequence: each finite simple planar graph has a vertex of [[degree|0T0F6CCS9W]] at most $5$.\n\n## Explanations\n\n### L1 human\n\nA planar graph cannot have too many edges: at most about three times its number of vertices.\nThis simple count already shows that some graphs, such as $K_5$, are not planar.\n",
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