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

2026-03-17

AI Summary

This study addresses the decision problem of whether a graph with maximum local edge-connectivity \(k\) is \(k\)-choosable. For 2-connected graphs, it establishes the first Brooks-type theorem in the list-coloring setting, providing a complete characterization: such a graph is \(k\)-choosable if and only if it does not belong to a specific graph class \(\mathcal{H}_k\). In the general case—including non-2-connected graphs—the paper proves that the decision problem is \(\Pi_2\)-complete, thereby revealing its high computational complexity. By integrating tools from graph theory, list coloring, Hajós constructions, and computational complexity theory, this work precisely delineates the boundary between structural properties and choosability.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionMachine Learning: Graph-based Machine Learning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Research challenges in human and human-AI computationSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 Hajós 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 $Π_2$-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.

Problem

• k-choosability
• maximum local edge-connectivity
• Brooks-type theorem
• graph coloring
• computational complexity

Innovation

• k-choosability
• maximum local edge-connectivity
• Brooks-type theorem
• Hajós join
• Π₂-completeness