  We define an extremal $(r|\chi)$-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 $(r|\chi)$-graphs and we exhibit several
$(r|\chi)$-graph constructions arising from Tur{\' a}n graphs.
