One more occurrence of variables makes satisfiability jump from trivial to NP-complete
SIAM Journal on Computing
On the bandwidth of triangulated triangles
Selected papers of the 14th British conference on Combinatorial conference
Journal of Graph Theory
Journal of Combinatorial Theory Series B
Efficient algorithms for Petersen's matching theorem
Journal of Algorithms
Journal of Combinatorial Theory Series B
Partitioning into graphs with only small components
Journal of Combinatorial Theory Series B
Bounded size components: partitions and transversals
Journal of Combinatorial Theory Series B
Relaxed two-coloring of cubic graphs
Journal of Combinatorial Theory Series B
Maximum acyclic and fragmented sets in regular graphs
Journal of Graph Theory
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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 (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C1 (C2, 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 ≤ C f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) ≤ 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 C1 = C2, of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.