In a certain planar embedding of a 3-regular connected graph $G$, the faces have degrees either $5$ or $6$. If there are $20$ faces of degree $6$ in the embedding, how many faces of degree $5$ are there?

I was planning on using the Handshaking Lemma for Faces equation but ran into the problem that I don't know how many edges there are in this graph.

Is Euler's Equation applicable in this situation even though we don't know how many vertices or edges there are?


Let $e$ be the number of edges, $v$ the number of vertices and $f$ the number of degree-$5$ faces. Then

  • by the handshaking lemma for vertices, $3v=2e$;
  • by the handshaking lemma for faces, $2e=120+5f$;
  • by Euler's formula, $(20+f)+v=e+2$.

Now solve.


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