LetG[H] be the lexicographic product and letG ⊠H be the strong product of the graphsG andH. It is proved that, ifG is aϰ-critical graph, then, for any graphH, $$\chi (G[H]) \leqslant \chi (H)(\chi (G) - 1) + \left[ {\frac{{\chi (H)}}{{\alpha (G)}}} \right].$$ This upper bound is used to calculate several chromatic numbers of strong products. It is shown in particular that fork⩾2, ϰ(C5 ⊠ C5 ⊠ C2k + 1) = 10 + ⌈5/k⌉, and fork ⩾ 2 andn ⩾ 1, $$\chi (\overline {C_{2k + 1} } )$$ ⊠K n ) =kn + ⌈n/2⌉. That the general upper bound cannot be improved for graphs which are notϰ-critical is demonstrated by two infinite series of graphs. The paper is concluded with an application to graph retracts: if for some graphH with at least one edgeϰ(G[H]) = ϰ(G)ϰ(H), then noϰ-critical subgraphG′ ofG, G′ ≠ K n, is a retract ofG.