Degrees and choice numbers

Abstract

The choice number ch(G) of a graph G = (V,E) is the minimum number k such that for every assignment of a list S(v) of at least k colors to each vertex v ∈ V , there is a proper vertex coloring of G assigning to each vertex v a color from its list S(v). We prove that if the minimum degree of G is d, then its choice number is at least ( 1 2 − o(1)) log2 d, where the o(1)-term tends to zero as d tends to infinity. This is tight up to a constant factor of 2 + o(1), improves an estimate established in [1], and settles a problem raised in [2].