The Chromatic Polynomial of a Digraph

TL;DR

This paper introduces a new way to compute the chromatic polynomial of a digraph using totally cyclic subdigraphs and poset structures, linking it to the underlying undirected graph.

Contribution

It presents a novel representation of the digraph chromatic polynomial via totally cyclic subdigraphs and decomposes the NL-coflow polynomial to relate it to the undirected case.

Findings

Representation of the polynomial using graded posets of cyclic subdigraphs

Decomposition of the NL-coflow polynomial

Equivalence of digraph and undirected graph chromatic polynomials for symmetric digraphs

Abstract

An acyclic coloring of a digraph as defined by 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, examining the special structure of the poset of totally cyclic subdigraphs with fixed underlying undirected graph. This decomposition 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.