  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.
