 
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 kchoosable 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 kchoosable.
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
