Is there a relation between Clique size $\omega(G)$ and genus $g(G)$ of a graph? That is does $$\omega(G)^c\geq g(G)\geq \omega(G)^{\frac{1}d}$$ hold with constants $c,d\geq1$?


migrated from cs.stackexchange.com Feb 10 '15 at 9:16

This question came from our site for students, researchers and practitioners of computer science.


No, we can have a graph with a constant clique number but an arbitrarily large graph genus.

The genus of the complete graph on $n \geq 3$ vertices $K_{n}$ is:

$$ \gamma(n) = \left\lceil\frac{(n-3)(n-4)}{12}\right\rceil $$

We can subdivide the edges of $K_{n}$ to obtain a graph $H$ with clique-number $\omega(H) = 2$ (or anything from $2$ to $n$ as we choose), but of course this still has a $K_{n}$ minor, and hence the same genus as $K_{n}$.


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.