Extremal regular graphs of given chromatic number

We define an extremal $(r\mathrm{\mid }\chi )$-graph as an $r$-regular graph with chromatic number $\chi$ of minimum order. We show that the Tur{' a}n graphs $T_{ak,k}$, the antihole graphs and the graphs $K_k\times K_2$ are extremal in this sense. We also study extremal Cayley $(r\mathrm{\mid }\chi )$-graphs and we exhibit several $(r\mathrm{\mid }\chi )$-graph constructions arising from Tur{' a}n graphs.