In a connected planar graph, every vertex has degree $3$, and every face is bordered by $5$ or $6$ edges. How many faces are bordered by $5$ edges?


Plug everything in the Euler's formula: $v-e+f=2$

$v=$#of vertices

$e=$#of edges

$f=$#of faces

$p=$#of pentagonal faces

$h=$#of hexagonal faces

So we have:

$e=\frac{5p+6h}{2}$ (each edge belongs exactly to two faces)

$v=\frac{5p+6h}{3}$ (each vertex is shared by exactly three faces)

$f=p+h$ (each face is either pentagonal or hexagonal)

Plugging into Euler's equation gives:


$h$ cancels out and the solution is $p=12$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.