Summary: Subgraphs with a Large Cochromatic Number
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.
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  and is related
to coloring problems and to Ramsey theory. The subject has been studied by various researches (see
 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