Acyclic dichromatic number of oriented graphs

TL;DR

This paper introduces the acyclic dichromatic number for directed graphs, explores its properties and differences from the dichromatic number, and establishes complexity results including NP-completeness for certain classes.

Contribution

It defines the new parameter acyclic dichromatic number, analyzes its properties, and compares it with the dichromatic number, including complexity results and generalizations.

Findings

Existence of digraphs with arbitrarily large difference between parameters

Deciding if acyclic dichromatic number ≤ 2 is NP-complete for bipartite digraphs

Polynomial-time decision for tournaments

Abstract

The dichromatic number $\vec{\chi}(D)$ of a digraph $D=(V,A)$ is the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ into $k$ subsets so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$. This is a generalization of the chromatic number for undirected graphs as a graph has chromatic number at most $k$ if and only if the complete biorientation of $G$ (replace each edge by a directed 2-cycle) has dichromatic number at most $k$. In this paper we introduce the acyclic dichromatic number $\vec{\chi}_{\rm a}(D)$ of a digraph $D$ as the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$ and each of the bipartite induced subdigraphs $D[V_i,V_j]$ is acyclic for each $1\leq i<j\leq k$. This parameter, which resembles the definition of acyclic chromatic number for undirected graphs, has apparently not been studied before. We derive a number of results which display the difference between the dichromatic number and the acyclic dichromatic number, in particular, there are digraphs $D$ with arbitrarily large $\vec{\chi}_{\rm a}(D)-\vec{\chi}(D)$, even among tournaments with dichromatic number 2 and bipartite tournaments (where the dichromatic number is always 2). We prove several complexity results, including that deciding whether $\vec{\chi}_{\rm a}(D)\leq 2$ is NP-complete already for bipartite digraphs, while it is polynomial for tournaments (contrary to the case for dichromatic number). We also generalize the concept of heroes of a tournament to acyclic heroes of tournaments.