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Suppose that $G$ is a graph with $n$ vertices and $5n−10$ edges. Prove that $G$ has a vertex of degree at most $9$.

I am stuck on how to approach this.

I see that there are two cases:

  1. $n \le 10$, in this case no vertex can have a degree higher than $9$.
  2. $n > 10$.

I cannot figure out how to prove the problem for the second case.

If it matters, we are working on planar graphs in class right now.

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Hint: The sum of the degrees is twice the number of edges, so $\sum \text{deg}(v) = 2(5n - 10) = 10n - 20$.

What happens if all of the vertices have degrees 10 or more?

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