Asymptotically the List Colouring Constants Are 1

In this paper we prove the following result about vertex list colourings, which shows that a conjecture of the first author (1999, J. Graph Theory31, 149–153) is asymptotically correct. Let G be a graph with the sets of lists S(v), satisfying that for every vertex ∣S(v)∣=(1+o(1))d and for each colour c∈S(v), the number of neighbours of v that have c in their list is at most d. Then there exists a proper colouring of G from these lists.