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Let $G = (V, E)$ be a graph with nine vertices, such that each vertex has degree $5$ or $6$. Show that $G$ has at least five vertices of degree $6$, or at least six vertices of degree $5$.

My friend and I have been working on this question for the past two days and we have nothing. Any pointers or help?

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If neither of those things are true, $G$ has exactly $5$ vertices of degree $5$ and exactly $4$ vertices of degree $6$. Can you see why this is impossible? – Micah May 11 '13 at 19:39
Oh my goodness, thank you. Both to Micah and Hagen. – Evan May 11 '13 at 19:59

Hint: The sum of all vertex degrees is twice the number of edges.

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