0
$\begingroup$

In each case, give the values of r, e, or v (whichever is not given) assuming that the graph is planar. Then either draw a connected, planar graph with the property, if possible, or explain why no such planar graph can exist.

I'm stuck on this one: 17 regions and every vertex has degree 5. Using Euler's formula, I believe it would be v - 34 + 17 = 2 with v = 19 which gives a valid answer because using 3V-6 theorem, 57 - 6 >= 34? But the answer is that it's not possible, why?

$\endgroup$
2
$\begingroup$

$v=19$ can't be true, since 5 is odd, and every graph has an even number of odd degree vertices by handshake theorem (Also note that $e=5v/2$ must be an integer)

We do know that $e=5v/2$, so we can plug this into Euler's formula and get: $$v - 5v/2+17=2$$ $$\implies v=10$$

So $v=10$. But since $G$ is assumed to be planar, we know that $e \leq 3v-6=24$. Since $G$ has 10 vertices, each with degree 5, $e=25$, so no such planar graph exists.

$\endgroup$

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.