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 Theory 31, 149-153) is asymptotically correct. Let G be a graph with the sets of lists S(υ), satisfying that for every vertex |S(υ)| = (1+o(1))d and for each colour c ∈ S(υ), the number of neighbours of υ that have c in their list is at most d. Then there exists a proper colouring of G from these lists.

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