Colouring graphs with constraints on connectivity

Abstract : A graph G has maximal local edge-connectivity k if the maximum number of edge-disjoint paths between every pair of distinct vertices x and y is at most k. We prove Brooks-type theorems for k-connected graphs with maximal local edge-connectivity k, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph G with maximal local connectivity 3, outputs an optimal colouring for G. On the other hand, we prove, for k ≥ 3, that k-colourability is NP-complete when restricted to minimally k-connected graphs, and 3-colourability is NP-complete when restricted to (k − 1)-connected graphs with maximal local connectivity k. Finally, we consider a parameterization of k-colourability based on the number of vertices of degree at least k + 1, and prove that, even when k is part of the input, the corresponding parameterized problem is FPT.
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Submitted on : Friday, July 28, 2017 - 10:56:09 AM
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Pierre Aboulker, Nick Brettell, Frédéric Havet, Dániel Marx, Nicolas Trotignon. Colouring graphs with constraints on connectivity. Journal of Graph Theory, Wiley, 2017, 85 (4), pp.814-838. ⟨10.1002/jgt.22109⟩. ⟨hal-01570035⟩

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