A simple, undirected graph $G=(V,E)$ is said to be *vertex-transitive* if for all $v,w\in V$ there is a graph isomorphism $\varphi:V\to V$ such that $\varphi(v) = w$.

The cyclic graphs $C_{2n+1}$ are vertex-transitive with chromatic number $3$.

Given any integers $k\geq 4$ and $n\geq k$, is there always a connected vertex-transitive graph $G=(V,E)$ with $\chi(G)=k$ and $|V|\geq n$?

vandw, but does involve free variablesaandb. $\endgroup$