Coloring [formula omitted]-partite sparse digraphs

Abstract

In this paper, we study colorings of k-partite sparse digraphs. The chromatic number of a graph G is the smallest integer k such that the vertices of G can be colored with k colors with the property that each color class is an independent set. The dichromatic number of a digraph D is the minimum k such that the vertices of D can be colored with k colors with each color class inducing an acyclic subdigraph. This coloring invariant shares many similarities with the graph chromatic number and can be thought of as its analogous digraph generalization.Our main result in this short note shows that there exist sparse k-partite digraphs which have dichromatic number k. This, in particular, not only implies that there exist graphs with equal chromatic and dichromatic number, but that they can be taken to be somewhat sparse.