Alon -- Tarsi numbers of direct products

TL;DR

This paper develops a framework for analyzing graph polynomials of Cartesian product graphs and applies it to establish new choosability bounds for certain product graphs based on their polynomial coefficients.

Contribution

It introduces a general framework for graph polynomial coefficients of Cartesian products and proves new choosability results for graphs with specific polynomial monomials.

Findings

If a graph's polynomial has an 'almost central' monomial, then its Cartesian product with an even cycle is (degree+2)-choosable.

The framework links polynomial coefficients to graph coloring properties.

New bounds on list coloring for product graphs based on polynomial structure.

Abstract

We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=(V,E)$ is a graph with degrees of vertices $2d(v), v\in V$, and the graph polynomial $\prod_{(i,j)\in E} (x_j-x_i)$ contains an "almost central" monomial (that means a monomial $\prod_v x_v^{c_v}$, where $|c_v-d(v)|\leqslant 1$ for all $v\in V$), then the Cartesian product $G\square C_{2n}$ is $(d(\cdot)+2)$-choosable.