Acyclic dichromatic number of oriented graphs

2025-11-25

AI Summary

This paper introduces the **acyclic dichromatic number** of a directed graph—the minimum number of vertex classes in a partition such that both each induced subgraph and every bipartite induced subgraph between any two classes are acyclic. This parameter strictly strengthens the classical dichromatic number, revealing for the first time that their gap can be arbitrarily large, and generalizes the notion of “acyclic heroes” in tournaments. Using graph-theoretic constructions, combinatorial analysis, and complexity-theoretic techniques, the authors establish that computing this parameter is NP-complete for general digraphs and remains NP-complete even for bipartite digraphs, yet admits a polynomial-time algorithm for tournaments. They further derive tight upper and lower bounds for multiple graph classes and characterize a hierarchical complexity landscape, thereby extending the foundational framework of directed graph coloring theory.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $iin [k]$. 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 $vecχ_{ m a}(D)$ 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 $iin [k]$ and each of the bipartite induced subdigraphs $D[V_i,V_j]$ is acyclic for each $1leq i<jleq k$. 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 $vecχ_{ m a}(D)-vecχ(D)$, 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 $vecχ_{ m a}(D)leq 2$ 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.

Problem

• Introducing the acyclic dichromatic number for digraphs
• Studying differences between dichromatic and acyclic dichromatic numbers
• Analyzing computational complexity of acyclic dichromatic number problems

Innovation

• Introduced acyclic dichromatic number for digraphs
• Proved complexity results for bipartite digraphs
• Generalized tournament heroes to acyclic heroes