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  "body": "\n## Proof\n\n1. Let $V$ be the vertex set and $E$ the edge set of $G$. Let $V_o$ be the set of vertices of odd [[degree|0T0F6CCS9W]], and $V_e = V \\setminus V_o$ the set of vertices of even degree.\n2. By the [[handshaking lemma|0XAP8WHNES]], $\\sum_{v \\in V} \\deg(v) = 2 \\abs{E}$, which is even.\n3. The sum $\\sum_{v \\in V_e} \\deg(v)$ is a sum of even numbers, so it is even.\n4. Hence $\\sum_{v \\in V_o} \\deg(v) = 2\\abs{E} - \\sum_{v \\in V_e} \\deg(v)$ is even.\n5. Each term of the sum in step 4 is odd, so it is congruent to $1$ modulo $2$. Thus that sum is congruent to $\\abs{V_o}$ modulo $2$.\n6. By steps 4 and 5, $\\abs{V_o}$ is even.\n\n## Idea\n\nThe total of all degrees is even. The even degrees do not change the parity, so the odd degrees must also add up to an even number, and that needs an even count of them.\n",
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