Acyclic dichromatic number of oriented graphs

The dichromatic number $\vecχ(D)$ 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 $\vecχ_{\rm a}(D)$ 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 $\vecχ_{\rm a}(D)-\vecχ(D)$, 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 $\vecχ_{\rm a}(D)\leq 2$ 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.