Alon–Krivelevich choosability conjecture for bipartite graphs

Let $G$ be a bipartite graph with maximum degree at most $\Delta$. Its choice number $\mathsf{ch}(G)$ is the least integer $q$ such that every list assignment giving each vertex at least $q$ colors admits a proper list coloring.

Alon–Krivelevich conjecture. There is an absolute constant $C$ such that

$$\mathsf{ch}(G) \leqslant C\log \Delta.$$

Equivalently, every bipartite graph of maximum degree at most $\Delta$ has choice number $O(\log \Delta)$. The conjecture is motivated by sharp results for complete and random bipartite graphs and remains open, although the best known general upper bounds are of order $\Delta/\log \Delta$.