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Deciding relaxed two-colourability: a hardness jump. (English) Zbl 1209.05071
Summary: We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph \(G\) is called \((C_{1}, C_{2})\)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most \(C_{1}(C_{2}\), resp.). We prove that the decision problem ‘Is there a (\(1, C\))-relaxed colouring of a given graph \(G\) of maximum degree 3?’ exhibits a hardness jump in the component order \(C\). In other words, there exists an integer \(f(3)\) such that the decision problem is NP-hard for every \(2 \leqslant C < f(3)\), while every graph of maximum degree 3 is \((1, f(3))\)-relaxed colourable. We also show \(f(3) \leqslant 22\) by way of a quasilinear time algorithm, which finds a (1,22)-relaxed colouring of any graph of maximum degree 3. Both the bound on \(f(3)\) and the running time greatly improve earlier results. We also study the symmetric version, that is, when \(C_{1} = C_{2}\), of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.

MSC:
05C15 Coloring of graphs and hypergraphs
05C85 Graph algorithms (graph-theoretic aspects)
68R10 Graph theory (including graph drawing) in computer science
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