Acyclic Coloring of Products of Digraphs and of Digraphs with Bounded Treewidth

Abstract

The dichromatic number of a digraph $G$ is the smallest integer $\chi_a(G)$ such that the vertex set of $G$ can be partitioned into $\chi_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of digraphs: $\chi_a(G\square H)=\max\{\chi_a(G),\chi_a(H)\}$; $\chi_a(G\times H)\le \min\{\chi_a(G),\chi_a(H)\}$; and $\chi_a(G[H]) = \chi_a(G[\overset{\leftrightarrow}{K}_k])$, where $k =\chi_a(H)$ and $\overset{\leftrightarrow}{K}_k$ denotes the complete digraph on $k$ vertices. In addition, we investigate the products of directed cycles, giving exact values for $\chi_a(\overset{\rightarrow}{C}_n\times \overset{\rightarrow}{C}_m)$ and $\chi_a(\overset{\rightarrow}{C}_n\boxtimes \overset{\rightarrow}{C}_m)$ for every $n,m$, and for $\chi_a(\overset{\rightarrow}{C}_n[H])$ for every positive integer $n$. This latter result generalizes a result given in \cite{PP.16}, where they give exact values when $n>\chi_a(H)$. We also provide a upper-bound to the dichromatic number of a digraph $G$ as a function of the treewidth of its underlying graph and we present an {\FPT}-time algorithm that computes the dichromatic number of $G$, when parameterized by treewidth of the underlying graph of $G$.