The chromatic polynomial of a digraph

Abstract

An acyclic coloring of a digraph as defined by V. Neumann-Lara is a vertex-coloring such that no monochromatic directed cycles occur. Counting the number of such colorings with k colors can be done by counting so-called Neumann-Lara-coflows (NL-coflows), which build a polynomial in k. We will present a representation of this polynomial using totally cyclic subdigraphs, which form a graded poset Q. Furthermore we will decompose our NL-coflow polynomial, which becomes the chromatic polynomial of a digraph by multiplication with the number of colors to the number of components, using the geometric structure of the face lattices of a class of polyhedra that corresponds to Q. This decomposition leads to a representation using certain subsets of edges of the underlying undirected graph and will confirm the equality of our chromatic polynomial of a digraph and the chromatic polynomial of the underlying undirected graph in the case of symmetric digraphs.