Exact Complexity of Exact-Four-Colorability

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 the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether χ(G) = 7, where DP = BH2(NP). Wagner raised the question of how small the numbers in a k-element set Mk can be chosen such that Exact-Mk-Colorability still is BH2k(NP)-complete. In particular, for k = 1, he asked if it is DP-complete to determine whether χ(G) = 4. In this note, we solve this question of Wagner and determine the precise threshold t ∈ {4, 5, 6, 7} for which the problem Exact-{t}-Colorability jumps from NP to DP-completeness: It is DP-complete to determine whether χ(G) = 4, yet Exact-{3}-Colorability is in NP. More generally, for each k ≥ 1, we show that Exact-Mk-Colorability is BH2k(NP)-complete for Mk = {3k + 1, 3k + 3, . . . , 5k − 1}. 1 Exact-Mk-Colorability and the Boolean Hierarchy over NP To classify the complexity of problems known to be NP-hard or coNP-hard, but seemingly not contained in NP ∪ coNP, Papadimitriou and Yannakakis [PY84] introduced DP, the class of differences of two NP problems. They showed that DP contains various interesting types of problems, including uniqueness problems, critical graph problems, and exact optimization problems. For example, Cai and Meyer [CM87] proved the DP-completeness of Minimal-3-Uncolorability, a critical graph problem that asks whether a given graph is not 3colorable, but deleting any of its vertices makes it 3-colorable. A graph is said to be k-colorable if its vertices can be colored using no more than k colors such that no two adjacent vertices receive ∗This work was supported in part by grant NSF-INT-9815095/DAAD-315-PPP-gü-ab and by a Heisenberg Fellowship of the Deutsche Forschungsgemeinschaft.