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