Colouring graphs with constraints on connectivity

TL;DR

This paper explores graph coloring under connectivity constraints, establishing Brooks-type theorems, polynomial algorithms for specific classes, and NP-completeness results for others, along with fixed-parameter tractability insights.

Contribution

It introduces new Brooks-type theorems for graphs with maximal local edge-connectivity and provides polynomial algorithms and complexity results for coloring such graphs.

Findings

Polynomial-time algorithm for 3-connected graphs with maximal local connectivity 3

NP-completeness of k-colorability for minimally k-connected graphs when k ≥ 3

Fixed-parameter tractability of k-colorability based on degree-related parameters

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 $k \ge 3$, 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.