A connected $k$-regular graph of order 12 is embedded in the plane, resulting in eight regions.

Question

A connected $k$-regular graph of order $12$ is embedded in the plane, resulting in eight regions. What is $k$?

Answer 1

Recall Euler's characteristic formula $f-e+v=2$ for a planar graph. So by the information given we have $8-e+12=2$. Then using the number of edges, the handshaking lemma, and the fact that the graph is $k$-regular, we can find $k$.