Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there a general way to define a lower bound on $|V(G)|$ given the girth $g(G)=g$ and chromatic number $\chi(G) = k$?

I heard there is a result, telling that $|V(G)| \geq k^{\frac{g}{2}}$, but I can't find it.

Thanks in advance.

share|improve this question

1 Answer 1

Some such lower bound may exist, but the inequality $\chi(G)^{g(G)} \leq n$ is very far from being true in general.

For one example, let $n \geq 3$ and let $G = K_n$. Then $\chi(G) = n$ and $g(G) = 3$, but $n^{3/2} > n$.

For another, let $n \geq 3$ and let $G = C_{2n}$. Then $\chi(G) = 2$ and $g(G) = 2n$, but $2^n > 2n$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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