Some extremal results in cochromatic and dichromatic theory

For a graph G , the cochromatic number of G , denoted z ( G ), is the least m for which there is a partition of the vertex set of G having order m . where each part induces a complete or empty graph. We show that if { G n } is a family of graphs where G n has o ( n 2 log 2 ( n )) edges, then z ( G n ) = o ( n ). We turn our attention to dichromatic numbers. Given a digraph D , the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G , the dichromatic number of G , denoted d ( G ), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d ( m ) we mean the minimum size of all graphs G where d ( G ) = m. We show that d ( m ) = θ( m 2 ln 2 ( m )).