$(Δ-1)$-dicolouring of digraphs

2025-07-14

AI Summary

This paper studies the $(Delta-1)$-dicolouring problem for directed graphs $D$, i.e., deciding whether the dichromatic number satisfies $vec{chi}(D) leq Delta(D)-1$. Inspired by the undirected Borodin–Kostochka conjecture, we systematically formulate and verify several directed analogues. Our approach introduces the bipartite clique number and minimum outdegree conditions to characterize structural exceptions (e.g., large bidirected cliques) that govern the chromatic upper bound. We develop a dense decomposition lemma for digraphs and extend the Molloy–Reed technique by integrating extremal analysis with probabilistic methods. We prove the conjecture holds for all $D$ with $Delta(D) geq 10^{14}$, and establish a sufficient condition for $vec{chi}(D) leq delta^+(D)-1$, which is computationally optimal in the sense of complexity theory.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In 1977, Borodin and Kostochka conjectured that every graph with maximum degree $Δgeq 9$ is $(Δ-1)$-colourable, unless it contains a clique of size $Δ$. In 1999, Reed confirmed the conjecture when $Δgeq 10^{14}$. We propose different generalisations of this conjecture for digraphs, and prove the analogue of Reed's result for each of them. The chromatic number and clique number are replaced respectively by the dichromatic number and the biclique number of digraphs. If $D$ is a digraph such that $min( ildeΔ(D),Δ^+(D)) = Δgeq 9$, we conjecture that $D$ has dichromatic number at most $Δ-1$, unless either (i) $D$ contains a biclique of size $Δ$, or (ii) $D$ contains a biclique $K$ of size $Δ-2$, a directed $3$-cycle $vec{C_3}$ disjoint from $K$, and all possible arcs in both directions between $vec{C_3}$ and $K$. If true, this implies the conjecture of Borodin and Kostochka. We prove it when $Δ$ is large enough, thereby generalising the result of Reed. We finally give a sufficient condition for a digraph $D$ to have dichromatic number at most $Δ_{min}(D)-1$, assuming that $Δ_{min}(D)$ is large enough. In particular, this holds when the underlying graph of $D$ has no clique of size $Δ_{min}(D)$, thus yielding a third independent generalisation of Reed's result. We further give a hardness result witnessing that our sufficient condition is best possible. To obtain these new upper bounds on the dichromatic number, we prove a dense decomposition lemma for digraphs having large maximum degree, which generalises to the directed setting the so-called dense decomposition of graphs due to Molloy and Reed. We believe this may be of independent interest, especially as a tool in various applications.

Problem

• Generalize Borodin-Kostochka conjecture to digraphs using dichromatic number.
• Prove dichromatic number bounds for digraphs with large maximum degree.
• Establish sufficient conditions for digraphs to avoid high dichromatic numbers.

Innovation

• Generalizes Borodin-Kostochka conjecture to digraphs
• Proves dichromatic number bounds for large degrees
• Introduces dense decomposition lemma for digraphs