Let $G=(V,E)$ be a finite, simple, undirected graph. A *dominating set* is a set $D\subseteq V$ such that for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$. The *dominating number* $\gamma(G)$ is defined to be the minimum cardinality of a dominating set.

The complete graphs show that the dominating number can be very small compared to the chromatic number $\chi(G)$. However, it feels like if $\chi(G)$ is "smallish" compared to $|V|$, then $\gamma(G)$ might have a chance to be larger than $\chi(G)$.

More mathematically: Is there $r\in ]0,1[$ such that whenever $\frac{\chi(G)}{|V(G)|} < r$ then $\gamma(G) \geq \chi(G)$?