Repositorio Institucional Académico. Universidad Andrés Bello. Chile

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Coloring Graphs with Constraints on Connectivity
(Wiley-Liss Inc., 2017-08) Aboulker, Pierre; Brettell, Nick; Havet, Frédéric; Marx, Dániel; Trotignon, Nicolas
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 coloring for G. On the other hand, we prove, for k≥3, that k-colorability is NP-complete when restricted to minimally k-connected graphs, and 3-colorability is NP-complete when restricted to (k − 3) -connected graphs with maximal local connectivity k. Finally, we consider a parameterization of k-colorability 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. © 2016 Wiley Periodicals, Inc.