A Brooks-type theorem for the k-choosability of graphs with maximum local edge-connectivity k

TL;DR

This paper extends Brooks-type theorems to k-choosability for graphs with maximum local edge-connectivity k, characterizing when such graphs are k-choosable and establishing complexity results for the decision problem.

Contribution

It provides a characterization of k-choosability for 2-connected graphs with maximum local edge-connectivity k, and proves the decision problem is a2_2-complete for general graphs with this property.

Findings

2-connected graphs with maximum local edge-connectivity k are k-choosable iff not in a4_k.

Deciding k-choosability in graphs with maximum local edge-connectivity k is a2_2-complete.

Generalizations of degree-choosability characterization are developed.

Abstract

For a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u,v)$-paths over all distinct pairs of vertices $(u,v)$ in $G$. Stiebitz and Toft (2018) proved a Brooks-type theorem for graphs with maximum local edge-connectivity $k$, showing that a graph with maximum local edge-connectivity $k$ is not $k$-colourable if and only if it has a block in $\mathcal{H}_k$, which is the class of graphs that can be obtained by taking Haj\'os joins of copies of $K_{k+1}$ and, when $k=3$, odd wheels. We prove that a $2$-connected graph with maximum local edge-connectivity $k$ is $k$-choosable if and only if it is not in $\mathcal{H}_k$. On the other hand, deciding $k$-choosability when restricted to graphs with maximum local edge-connectivity $k$ (that might not be $2$-connected) is $\Pi_2$-complete. To prove the former result, we first prove several generalisations of a well-known characterisation of degree-choosability; these may be of independent interest.