Exact complexity of Exact-Four-Colorability

Abstract

Let M k be a given set of k integers. Define Exact-M k -Colorability to be the problem of determining whether or not χ(G), the chromatic number of a given graph G, equals one of the k elements of the set M k exactly. In 1987, Wagner [Theoret. Comput. Sci. 51 (1987 proved that Exact-M k -Colorability is BH 2k (NP)-complete, where M k = {6k + 1, 6k + 3, . . . , 8k − 1} and BH 2k (NP) is the 2kth level of the Boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not χ(G) = 7, where DP = BH 2 (NP). Wagner raised the question of how small the numbers in a k-element set M k can be chosen such that Exact-M k -Colorability still is BH 2k (NP)-complete. In particular, for k = 1, he asked if it is DP-complete to determine whether or not χ(G) = 4.In this note, we solve Wagner's question and prove the optimal result: For each k 1, Exact-M k -Colorability is BH 2k (NP)-complete for M k = {3k + 1, 3k + 3, . . . , 5k − 1}. In particular, for k = 1, we determine the precise threshold of the parameter 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 or not χ(G) = 4, yet Exact-{3}-Colorability is in NP. troduced 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 [7] proved the DPcompleteness of Minimal-3-Uncolorability, a critical graph problem that asks whether a given graph is not 3-colorable, 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 the same color. The chromatic number of a graph G, denoted χ(G), is defined to be the smallest k such that G is k-colorable.

Satisfiability Parsimoniously Reduces to the Tantrix™ Rotation Puzzle Problem

“…Note that DP is the second level of the boolean hierarchy over NP, see Cai et al [3,4]. Further results on DP and completeness in the boolean hierarchy over NP can be found, e.g., in [5,15], see also the survey [14].…”

QDEF and Its Approximations in OBDM

“…Finally, as for the perfect case, the proof of DP-hardness is by a LOGSPACE reduction from the exact-4-colourability problem, which is DP-complete [33]. In particular, a graph G is exact-4-colourable (i.e., 4-colourable and not 3-colourable) if and only if cert D qG,J = {(c 4 )}, i.e., if and only if cert D qG,J = λ.…”

On computing the smallest four-coloring of planar graphs and non-self-reducible sets in P

“…Papadimitriou [17] proved that Unique-Optimal-Traveling-Salesperson is Δ p 2 -complete, and Krentel [14] and Wagner [27] lished many more Δ p 2 -completeness results, including the result that the problem Odd-Max-SAT is Δ p 2 -complete. The complexity of colorability problems has been studied in a number of papers, see, e.g., [1,2,6,12,19,24,27].…”