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

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.