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Deciding Relaxed Two-Colorability—A Hardness Jump

  • Robert Berke
  • Tibor Szabó
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)

Abstract

A coloring is proper if each color class induces connected components of order one (where the order of a graph is its number of vertices). Here we study relaxations of proper two-colorings, such that the order of the induced monochromatic components in one (or both) of the color classes is bounded by a constant. In a (C 1 , C 2 )-relaxed coloring of a graph G every monochromatic component induced by vertices of the first (second) color is of order at most C 1 (C 2, resp.). We are mostly concerned with (1,C)-relaxed colorings, in other words when/how is it possible to break up a graph into small components with the removal of an independent set.

We prove that every graph of maximum degree at most three can be (1,22)-relaxed colored and we give a quasilinear algorithm which constructs such a coloring. We also show that a similar statement cannot be true for graphs of maximum degree at most 4 in a very strong sense: we construct 4-regular graphs such that the removal of any independent set leaves a connected component whose order is linear in the number of vertices.

Furthermore we investigate the complexity of the decision problem (Δ, C)-AsymRelCol: Given a graph of maximum degree at most Δ, is there a (1,C)-relaxed coloring of G? We find a remarkable hardness jump in the behavior of this problem. We note that there is not even an obvious monotonicity in the hardness of the problem as C grows, i.e. the hardness for component order C+1 does not imply directly the hardness for C. In fact for C=1 the problem is obviously polynomial-time decidable, while it is shown that it is NP-hard for C=2 and Δ≥3.

For arbitrary Δ≥2 we still establish the monotonicity of hardness of (Δ, C)-AsymRelCol on the interval 2≤C ≤∞ in the following strong sense. There exists a critical component order f(Δ)∈ℕ∪{∞} such that the problem of deciding (1,C)-relaxed colorability of graphs of maximum degree at most Δ is NP-complete for every 2≤C< f(Δ), while deciding (1,f(Δ))-colorability is trivial: every graph of maximum degree Δ is (1,f(Δ))-colorable. For Δ=3 the existence of this threshold is shown despite the fact that we do not know its precise value, only 6≤f(3)≤22. For any Δ≥4, (Δ, C)-AsymRelCol is NP-complete for arbitrary C≥2, so f(Δ)=∞.

We also study the symmetric version of the relaxed coloring problem, and make the first steps towards establishing a similar hardness jump.

Keywords

Perfect Match Maximum Degree Chromatic Number Conjunctive Normal Form Full Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Robert Berke
    • 1
  • Tibor Szabó
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichSwitzerland

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