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List coloring of random and pseudo-random graphs Michael Krivelevich

Summary: List coloring of random and pseudo-random graphs
Noga Alon
Michael Krivelevich
Benny Sudakov
The choice number of a graph G is the minimum integer k such that for every assignment of
a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each
vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is
almost surely ( np(n)
ln(np(n)) ) whenever 2 < np(n) n/2. A related result for pseudo-random graphs
is proved as well. By a special case of this result, the choice number (as well as the chromatic
number) of any graph on n vertices with minimum degree at least n/2 - n0.99
in which no two
distinct vertices have more than n/4 + n0.99
common neighbors is at most O(n/ ln n).
1 Introduction
A vertex-coloring of a graph G is an assignment of a color to each of its vertices. The coloring
is proper if no two adjacent vertices get the same color. The chromatic number (G) of G is the
minimum number of colors used in a proper coloring of it. If (G) k we say that G is k-colorable.
A related, more complicated quantity is the choice number ch(G) of G, introduced in [11] and


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics