Coloring digraphs with $Δ-b$ colors

2026-07-07

AI Summary

This study addresses the acyclic vertex coloring of directed graphs, aiming to minimize the number of colors such that each color class induces an acyclic subgraph, and uncovers its intrinsic connection to complete bipartite subdigraphs (bicliques). Inspired by Reed’s classical work on undirected graphs, the authors extend the theory to digraphs by introducing the parameter Δ(D), defined as the maximum square root of the product of a vertex’s in-degree and out-degree. They prove that for any fixed integer b, every digraph with sufficiently large Δ either contains a biclique of order exceeding Δ − 2b or has dichromatic number at most Δ − b. This result confirms a conjecture previously posed by the authors, and the tightness of the bound is demonstrated through explicit constructions.

Abstract

The dichromatic number of a digraph is the minimum number of colors needed to partition its vertex set into acyclic subdigraphs. A biclique is a set of vertices inducing all possible pairs of opposite arcs. For a digraph $D$, define $Δ(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We prove that, for every fixed integer $b\in\mathbb{N}$, every digraph $D$ with $Δ(D) = Δ$ being sufficiently large with respect to $b$ either contains a biclique whose size exceeds $Δ-2b$ or has dichromatic number at most $Δ-b$. This extends a classical result of Reed to the directed setting and supports a conjecture of the present authors. Furthermore, the theorem is tight, as for all integers $b$ and $Δ\geq 3b$ there exists a digraph $D$ with $Δ(D)= Δ$, dichromatic number $Δ-b+1$, and whose largest biclique has size $Δ-2b+1$.

Problem

• dichromatic number
• digraph
• biclique
• coloring
• maximum degree

Innovation

• dichromatic number
• biclique
• directed graph
• graph coloring
• degree condition