Strengthening the Directed Brooks' Theorem for oriented graphs and consequences on digraph redicolouring

TL;DR

This paper refines the understanding of the dichromatic number in digraphs, establishing new bounds and structural properties, and explores recoloring dynamics and the structure of the dicoloring graph.

Contribution

It strengthens the directed Brooks' theorem for oriented graphs, characterizes extremal cases, and extends recoloring and connectivity results to digraphs.

Findings

Dichromatic number equals ((D))+1 only in specific join structures.

Oriented graphs with ((G))  2 have dichromatic number at most ((G)).

Recoloring between two 2-dicolourings can be done in at most n steps.

Abstract

Let $D=(V,A)$ be a digraph. We define $\Delta_{\max}(D)$ as the maximum of $\{ \max(d^+(v),d^-(v)) \mid v \in V \}$ and $\Delta_{\min}(D)$ as the maximum of $\{ \min(d^+(v),d^-(v)) \mid v \in V \}$. It is known that the dichromatic number of $D$ is at most $\Delta_{\min}(D) + 1$. In this work, we prove that every digraph $D$ which has dichromatic number exactly $\Delta_{\min}(D) + 1$ must contain the directed join of $\overleftrightarrow{K_r}$ and $\overleftrightarrow{K_s}$ for some $r,s$ such that $r+s = \Delta_{\min}(D) + 1$, except if $\Delta_{\min}(D) = 2$ in which case $D$ must contain a digon. In particular, every oriented graph $\vec{G}$ with $\Delta_{\min}(\vec{G}) \geq 2$ has dichromatic number at most $\Delta_{\min}(\vec{G})$. Let $\vec{G}$ be an oriented graph of order $n$ such that $\Delta_{\min}(\vec{G}) \leq 1$. Given two 2-dicolourings of $\vec{G}$, we show that we can transform one into the other in at most $n$ steps, by recolouring one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph $\vec{G}$ on $n$ vertices, the distance between two $k$-dicolourings is at most $2\Delta_{\min}(\vec{G})n$ when $k\geq \Delta_{\min}(\vec{G}) + 1$. We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph $D$ with $\Delta_{\max}(D) = \Delta \geq 3$ and every $k\geq \Delta +1$, the $k$-dicolouring graph of $D$ consists of isolated vertices and at most one further component that has diameter at most $c_{\Delta}n^2$, where $c_{\Delta} = O(\Delta^2)$ is a constant depending only on $\Delta$.