Is there a low degree vertex in planar graphs? A book I read states "Since $G$ is planar, $G$ has a vertex $v$ of degree at most 5".


Is this true?
Why?
Does this theorem have a name?
Yes, this is true.
It is a direct consequence of Euler's formula. The formula entails that the average degree is strictly less than 6, which implies the existence of a vertex of degree at most 5.
It doesn't have a name in itself, but Euler's formula is the overall subject.