On two generalizations of the Alon-Tarsi polynomial method

TL;DR

This paper extends the algebraic Alon-Tarsi polynomial method to broader graph classes, providing new characterizations and proving equality of chromatic and choice numbers with the Alon-Tarsi number for specific graph families.

Contribution

It generalizes the Alon-Tarsi characterizations using multiple weight functions and extends these results to all graphs and hypergraphs.

Findings

Generalized Alon-Tarsi characterizations with multiple weight functions

Proved equality of chromatic, list chromatic, and Alon-Tarsi numbers for certain graph families

Extended results to hypergraphs and broader graph classes

Abstract

In a seminal paper, Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of $G$ obtained via their method, was later coined the \emph{Alon-Tarsi number of $G$} and was denoted by $AT(G)$. They have provided a combinatorial interpretation of this parameter in terms of the eulerian subdigraphs of an appropriate orientation of $G$. Their characterization can be restated as follows. Let $D$ be an orientation of $G$. Assign a weight $\omega_D(H)$ to every subdigraph $H$ of $D$: if $H \subseteq D$ is eulerian, then $\omega_D(H) = (-1)^{e(H)}$, otherwise $\omega_D(H) = 0$. Alon and Tarsi proved that $AT(G) \leq k$ if and only if there exists an orientation $D$ of $G$ in which the out-degree of every vertex is strictly less than $k$, and moreover $\sum_{H \subseteq D} \omega_D(H) \neq 0$. Shortly afterwards, for the special case of line graphs of $d$-regular $d$-edge-colorable graphs, Alon gave another interpretation of $AT(G)$, this time in terms of the signed $d$-colorings of the line graph. In this paper we generalize both results. The first characterization is generalized by showing that there is an infinite family of weight functions (which includes the one considered by Alon and Tarsi), each of which can be used to characterize $AT(G)$. The second characterization is generalized to all graphs (in fact the result is even more general -- in particular it applies to hypergraphs). We then use the second generalization to prove that $\chi(G) = ch(G) = AT(G)$ holds for certain families of graphs $G$. Some of these results generalize certain known choosability results.