[math-fun] non-orientable chromaticity formula?

Thanks for all your interesting responses to my query about fractional genera! So... what about the more specific question: is there a known formula that gives the chromatic number of a surface with x cross-caps like there is for one with g holes? It should give 4 with x=0 and (I think) 6 for x=1, right? (Substituting g=x/2 in the orientable formula gives these values, which is what triggered the more general flight of fancy.) What about x=2?