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