 
Summary: Subgraphs with a Large Cochromatic Number
Noga Alon
Michael Krivelevich
Benny Sudakov
Abstract
The cochromatic number of a graph G = (V, E) is the smallest number of parts in a partition
of V in which each part is either an independent set or induces a complete subgraph. We show
that if the chromatic number of G is n, then G contains a subgraph with cochromatic number at
least ( n
ln n ). This is tight, up to the constant factor, and settles a problem of Erdos and Gimbel.
1 Introduction
All graphs considered here are finite and simple. For a graph G, let (G) denote the chromatic
number of G. The cochromatic number of G = (V, E) is the smallest number of sets into which the
vertex set V can be partitioned so that each set is either independent or induces a complete graph.
We denote by z(G) the cochromatic number of G.
The cochromatic number was originally introduced by L. Lesniak and H. Straight [6] and is related
to coloring problems and to Ramsey theory. The subject has been studied by various researches (see
[8] for several references). A natural question is to find a connection between the chromatic and
the cochromatic numbers of a graph. A complete graph on n vertices shows that a graph G with
a high chromatic number may have a low cochromatic number. Thus to get a nontrivial result one
