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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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##### References:
 [1] DOI: 10.1006/jctb.2000.2018 · Zbl 1028.05054 · doi:10.1006/jctb.2000.2018 [2] DOI: 10.1006/jctb.1998.1868 · Zbl 0930.05043 · doi:10.1006/jctb.1998.1868 [3] DOI: 10.1002/jgt.3190100207 · Zbl 0596.05024 · doi:10.1002/jgt.3190100207 [4] DOI: 10.1016/j.jctb.2006.12.001 · Zbl 1118.05029 · doi:10.1016/j.jctb.2006.12.001 [5] DOI: 10.1007/BF01303516 · Zbl 0790.05067 · doi:10.1007/BF01303516 [6] DOI: 10.1137/0222015 · Zbl 0767.68057 · doi:10.1137/0222015 [7] DOI: 10.1016/S0095-8956(02)00006-0 · Zbl 1023.05045 · doi:10.1016/S0095-8956(02)00006-0 [8] DOI: 10.1016/0012-365X(94)00208-Z · Zbl 0823.05048 · doi:10.1016/0012-365X(94)00208-Z [9] Akiyama, Bull. Liber. Arts Sci. NMS 2 pp 1– (1981) [10] DOI: 10.1002/jgt.20271 · Zbl 1131.05050 · doi:10.1002/jgt.20271 [11] DOI: 10.1016/S0095-8956(03)00031-5 · Zbl 1033.05083 · doi:10.1016/S0095-8956(03)00031-5 [12] Harary, Graphs and Applications pp 127– (1985) [13] DOI: 10.1016/0166-218X(84)90081-7 · Zbl 0534.68028 · doi:10.1016/0166-218X(84)90081-7 [14] DOI: 10.1006/jagm.2000.1132 · Zbl 0969.68179 · doi:10.1006/jagm.2000.1132 [15] DOI: 10.1007/BF01202473 · Zbl 0796.05036 · doi:10.1007/BF01202473 [16] DOI: 10.1002/(SICI)1097-0118(199703)24:33.0.CO;2-T · doi:10.1002/(SICI)1097-0118(199703)24:33.0.CO;2-T
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