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  "body": "\n## Statement\n\nLet $G$ be a finite [[graph|0JD5C82QMB]], and let $W$ be an [[Eulerian trail|0NNYDD1AVR]] of $G$ from $u$ to $v$.\nIf $u = v$, then every vertex of $G$ has even [[degree|0T0F6CCS9W]].\nIf $u \\neq v$, then $u$ and $v$ have odd degree and every other vertex has even degree.\n\n## Remarks\n\nProof idea: each time $W$ passes through a vertex, it uses two edges at that vertex. Only the start and the end use one edge without a partner.\nThus a graph with an Eulerian trail has $0$ or $2$ vertices of odd degree. The [[Seven Bridges of Königsberg|0YCW781Q9D]] have $4$.\n\n## Explanations\n\n### L1 human\n\nCount the edges at a vertex as a walk uses them.\nEvery visit in the middle of the walk uses one edge in and one edge out.\nOnly the first and last vertices can be left with an odd count.\n",
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