Being new to the real of Graph theory, I came across this problem where it was asked to find the number of regions in the planar depiction of G, where G=(V,E), loop-free, connected 4-regular planar graph, where |E| = $e$ = 16
The way I figured I had to solve this problem, was to use the handshaking lemma for undirected graphs in order to find the |V| = $v$ and subsequently use the value of $v$ in the Euler's theorem: $v - e + r=2$ to solve for $r$.
Any suggestions for other methods for solving this problem?
A 4-regular graph with 16 edges has 8 vertices, since $(|V| * 4) / 2 = 16$. According to Euler's formula it has 10 faces, which agrees with the planar depiction of the 4-antiprism graph for example.