List colorings of direct products

Let $G=C_n\times C_m$ be a toroidal grid (that is, 4-regular graph), where $nm$ is even. We prove that this graph $G$ is 3-choosable. We also prove some more general results about list colorings of direct products. The proofs are algebraic, the starting point is Alon — Tarsi application of Combinatorial Nullstellensatz, and the main difficulty is to prove that the corresponding coefficient of the graph polynomial is non-zero.

The talk is based on joint results with Alexey Gordeev, Zhiguo Li and Zeling Shao.

No preliminary knowledge is required.