Deciding relaxed two-colorability - A hardness jump

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 (C1, C2)-relaxed coloring of a graph G every monochromatic component induced by vertices of the first (second) color is of order at most C1 (C2, 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. © Springer-Verlag Berlin Heidelberg 2006.