Colouring graphs with constraints on connectivity

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.