DeepAI

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

Let D=(V,A) be a digraph. We define Δ_max(D) as the maximum of {max(d^+(v),d^-(v)) | v ∈ V } and Δ_min(D) as the maximum of {min(d^+(v),d^-(v)) | v ∈ V }. It is known that the dichromatic number of D is at most Δ_min(D) + 1. In this work, we prove that every digraph D which has dichromatic number exactly Δ_min(D) + 1 must contain the directed join of K_r and K_s for some r,s such that r+s = Δ_min(D) + 1. In particular, every oriented graph G⃗ with Δ_min(G⃗) ≥ 2 has dichromatic number at most Δ_min(G⃗). Let G⃗ be an oriented graph of order n such that Δ_min(G⃗) ≤ 1. Given two 2-dicolourings of G⃗, 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 G⃗ on n vertices, the distance between two k-dicolourings is at most 2Δ_min(G⃗)n when k≥Δ_min(G⃗) + 1. We then extend a theorem of Feghali to digraphs. We prove that, for every digraph D with Δ_max(D) = Δ≥ 3 and every k≥Δ +1, the k-dicolouring graph of D consists of isolated vertices and at most one further component that has diameter at most c_Δn^2, where c_Δ = O(Δ^2) is a constant depending only on Δ.

01/23/2015

A Reconfigurations Analogue of Brooks' Theorem and its Consequences

Let G be a simple undirected graph on n vertices with maximum degree Δ. ...
01/09/2023

Digraph redicolouring

Given two k-dicolourings of a digraph D, we prove that it is PSPACE-comp...
03/13/2019

A polynomial version of Cereceda's conjecture

Let k and d be such that k > d+2. Consider two k-colourings of a d-degen...
12/05/2022

(P6, triangle)-free digraphs have bounded dichromatic number

The dichromatic number of an oriented graph is the minimum size of a par...
10/04/2019

To reorient is easier than to orient: an on-line algorithm for reorientation of graphs

We define an on-line (incremental) algorithm that, given a (possibly inf...
04/29/2019

The graphs behind Reuleaux polyhedra

This work is about graphs arising from Reuleaux polyhedra. Such graphs m...
11/25/2021

On Queries Determined by a Constant Number of Homomorphism Counts

It is well known [Lovász, 1967] that up to isomorphism a graph G is dete...