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?
Is there a low degree vertex in planar graphs?
graph theory
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.