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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.

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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 $2^n > 2n$.

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