An analogue of Reed's conjecture for digraphs

TL;DR

This paper proposes a conjecture for the chromatic number of digraphs analogous to Reed's conjecture for graphs, providing partial results and bounds that extend known results to directed graphs.

Contribution

It introduces a new conjecture for digraphs related to their dichromatic number, and proves partial bounds and generalizations of existing results for directed graphs.

Findings

Established an upper bound on the dichromatic number involving maximum degree and biclique size.

Proved that digraphs with large bicliques admit acyclic sets intersecting all bicliques.

Improved bounds on the dichromatic number for oriented graphs.

Abstract

Reed in 1998 conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil \frac{\Delta(G)+1+\omega(G)}{2} \rceil$. As a partial result, he proved the existence of $\varepsilon > 0$ for which every graph $G$ satisfies $\chi(G) \leq \lceil (1-\varepsilon)(\Delta(G)+1)+\varepsilon\omega(G) \rceil$. We propose an analogue conjecture for digraphs. Given a digraph $D$, we denote by $\vec{\chi}(D)$ the dichromatic number of $D$, which is the minimum number of colours needed to partition $D$ into acyclic induced subdigraphs. We let $\overleftrightarrow{\omega}(D)$ denote the size of the largest biclique (a set of vertices inducing a complete digraph) of $D$ and $\tilde{\Delta}(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We conjecture that every digraph $D$ satisfies $\vec{\chi}(D) \leq \lceil \frac{\tilde{\Delta}(D)+1+\overleftrightarrow{\omega}(D)}{2} \rceil$, which if true implies Reed's conjecture. As a partial result, we prove the existence of $\varepsilon >0$ for which every digraph $D$ satisfies $\vec{\chi}(D) \leq \lceil (1-\varepsilon)(\tilde{\Delta}(D)+1)+\varepsilon\overleftrightarrow{\omega}(D) \rceil$. This implies both Reed's result and an independent result of Harutyunyan and Mohar for oriented graphs. To obtain this upper bound on $\vec{\chi}$, we prove that every digraph $D$ with $\overleftrightarrow{\omega}(D) > \frac{2}{3}(\Delta_{\max}(D)+1)$, where $\Delta_{\max}(D) = \max_{v\in V(D)} \max(d^+(v),d^-(v))$, admits an acyclic set of vertices intersecting each biclique of $D$, which generalises a result of King. We finally give a short proof that all oriented graphs $D$ satisfy $\vec{\chi}(D) \leq \frac{\sqrt{2}}{2} \Tilde{\Delta}(D) + 2$, improving on a result of Golowich.