Degrees and choice numbers The choice number ch(G) of a graph G = (V, E) is the minimum number k such that for every Summary: Degrees and choice numbers Noga Alon 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]. 1 Introduction An undirected, simple graph G = (V, E) is k-choosable if 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). The choice number ch(G) of G, (which is also called the list chromatic number of G) is the minimum number k such that G is k-choosable. The concept of choosability, introduced by Vizing in 1976 [6] and independently by Erdos, Rubin and Taylor in 1979 [4], received a considerable amount of attention recently. Many of the recent results can be found in the survey papers [1], [5] and their many references. By definition, the choice number ch(G) of any graph G is at least as large as its chromatic number (G), and it is well known that strict inequality can hold. In fact, it is shown in [4] that the choice number of the complete bipartite Collections: Mathematics