An efficient implementation and a strengthening of Alon-Tarsi list coloring method

TL;DR

This paper presents an efficient implementation of the Alon-Tarsi list coloring method, enhancing its practical applicability and providing new constraints to determine graph choosability.

Contribution

It introduces a computationally feasible implementation of the Alon-Tarsi method and extends its capability to confirm colorability using additional polynomial coefficients.

Findings

Feasible testing of graph choosability for graphs with around 70 edges.

Additional polynomial coefficients provide constraints that help confirm or deny list colorability.

Implementation available at https://gitlab.mff.cuni.cz/dvorz9am/alon-tarsi-method.

Abstract

As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi gave a way to prove that a graph is choosable (colorable from any lists of prescribed size). We describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges. We also show that in case that Alon-Tarsi method fails to show that the graph is choosable, further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. The implementation can be found at https://gitlab.mff.cuni.cz/dvorz9am/alon-tarsi-method.