Exact complexity of exact-four-colorability

  • Authors:

  • Jörg Rothe

  • Affiliations:

  • Abteilung für Informatik, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany

  • Venue:
  • Information Processing Letters
  • Year:
  • 2003

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Abstract

Let Mk be a given set of k integers. Define Exact-Mk-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 Mk exactly. In 1987, Wagner [Theoret. Comput. Sci. 51 (1987) 53-80] 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 or not 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 or not χ(G) = 4.In this note, we solve Wagner's question and prove the optimal result: For each k ≥ 1, Exact-Mk-Colorability is BH2k(NP)-complete for Mk = {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.