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I just need help verifying if my solution is correct or not.
2026-03-29 15:32:46.1774798366
Prove That Every Simple Graph Has Two Vertices Of The Same Degree.
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Your proof looks fine. I could probably be simplified a little bit, but it's fine.
In particular, the last paragraph is redundant, and can be fixed by slightly altering the end of the 2nd-to-last:
"...there must be a vertex $A$ of degree 0, and a different vertex $B$ of degree $n-1$. Since there are exactly $n-1$ non-$B$ vertices, and the graph is simple, $B$ must share an edge with every other vertex, including $A$. But that contradicts the fact that $\deg A = 0$."