Acyclic dichromatic number of oriented graphs

TL;DR

We study the acyclic dichromatic number $\vec{\chi}_{\rm a}(D)$, a directed analogue of the acyclic chromatic number requiring each color class to induce an acyclic subdigraph and all bipartite color-class subdigraphs to be acyclic. We show that $\vec{\chi}_{\rm a}(D)$ can be arbitrarily larger than the ordinary dichromatic number $\vec{\chi}(D)$, even for tournaments with $\vec{\chi}(D)=2$, via a vertex-splitting construction that yields $\vec{\chi}_{\rm a}(D(G))=\lceil\sqrt{\chi(G)}\rceil$. The paper establishes NP-hardness of deciding $\vec{\chi}_{\rm a}(D)\le k$ for fixed $k$, including split digraphs, while showing a polynomial-time algorithm for the case $\vec{\chi}_{\rm a}(T)\le 2$ in tournaments. It introduces acyclic heroes of tournaments and proves several results—most notably that $\Delta(k,1,1)$ is an acyclic hero—along with conjectures aiming to classify all acyclic heroes. Additionally, tight bounds are derived for $\vec{\chi}_{\rm a}$ in tournaments, and the behavior on degenerate graphs is explored, together with open problems and directions for future research.

Abstract

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.

Figures (5)

• Figure 1: Figures (b) and (d) illustrate the digraph $D(G)$ for the two graphs shown in Figures (a) and (c), respectively.
• Figure 2: A digraph $D$ together with one of its $2$-dicolourings $\varphi$ (left), and the corresponding digraph $D^\star$ with one of its acyclic $3$-dicolourings (right).
• Figure 3: The partition $(V(P),X_0,\dots,X_{\ell+1})$.
• Figure 4: The only possible structure of any $0$-coherent colouring of $T[Z_0]$. Vertices coloured $1$ are in blue, and vertices coloured $2$ in orange. As indicated by the twisting arrows, the acyclic ordering of each $X_j$, $j\geqslant 1$, goes from right to left.
• Figure 5: The digraph $W[x,y]$ together with one of its acyclic $2$-dicolourings.

Theorems & Definitions (68)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 4: Berger et al. bergerJCT103
• Conjecture 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9: Harutyunyan et al. harutyunyanJTCB138