**Conjecture**There is a constant such that the list chromatic number of any bipartite graph of maximum degree is at most .

For definitions and an introduction to list colouring, see the related Wikipedia page.

Alon [A] showed that the list chromatic number of a graph (not necessarily bipartite) of maximum degree is at least . Random bipartite graphs show that this is tight up to a multiplicative factor .

It is not diffcult to see that the list chromatic number of any bipartite graph of maximum degree is at most . It also follows a more general result of Johansson [J] on triangle-free graphs.

## Bibliography

*[A] N. Alon, Degrees and choice numbers, Random Structures Algorithms, 16 (2000), 364--368.

[AK] N. Alon and M. Krivelevich, The choice number of random bipartite graphs, Annals of Combi- natorics 2 (1998), 291-297.

[J] A. Johansson. Asymptotic choice number for triangle free graphs. Technical Report 91–95, DIMACS, 1996.

* indicates original appearance(s) of problem.