  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.
