# 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 *(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## Preview

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