Graph colorings and recursively bounded Π10-classes

Abstract Bean showed that the k-colorings of a recursive graph is always a recursively bounded Π10-class. The main theorem of this paper proves the converse of Bean's result. That is, we show that for any recursively bounded Π10-class C and any k ⩾ 3, there exists a highly recursive graph G such that up to a permutation of colors there is an effective 1:1 degree preserving correspondence between the k-colorings of G and the elements of C . We also discuss other combinatorial problems whose solutions are recursively bounded Π10-classes and the possibility of their representing an arbitrary Π10-class.