Let Mk ⊆ N be a given set that consists of k noncontiguous integers. Define Exact-Mk-Colorability to be the problem of determining whether χ(G), the chromatic number of a given graph G, equals one of the k elements of the set Mk exactly. In 1987, Wagner [Wag87] proved that Exact-Mk-Colorability is BH2k(NP)-complete, where Mk = {6k + 1, 6k + 3, . . . , 8k − 1} and BH2k(NP) is the 2kth level of t...