In this work, we generalize several results on graph recolouring to digraphs. Given two k-dicolourings of a digraph D, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for k = 2 and for digraphs with maximum degree 5 or oriented planar graphs with maximum degree 6.A digraph is said to be k-mixing if there exists a transformation between any pair of k-dicolourings. We show that every digraph D is k-mixing for all k ≥ δ * min (D) + 2, generalizing a result due to Dyer et al. We also prove that every oriented graph G is k-mixing for all k ≥ δ * max ( G) + 1 and for all k ≥ δ * avg ( G) + 1. Here δ * min , δ * max , and δ * avg denote the min-degeneracy, the max-degeneracy, and the average-degeneracy respectively. We pose as a conjecture that, for every digraph D, the dicolouring graph of D on k ≥ δ * min (D) + 2 colours has diameter at most O(|V (D)| 2 ). This is the analogue of Cereceda's conjecture for digraphs. We generalize to digraphs two results supporting Cereceda's conjecture. We first prove that the dicolouring graph of any digraph D on k ≥ 2δ * min (D) + 2 colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the analogue of Cereceda's conjecture is true when k ≥ 3 2 (δ * min (D) + 1), which generalizes a result from Bousquet and Heinrich.Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on k ≥ 2 colours is connected and has diameter at most 2n. We conjecture that every non 2-mixing oriented graph has maximum average degree at least 4, and we provide some support for this conjecture by proving it on the special case of 2-freezable oriented graphs. More generally, we show that every k-freezable oriented graph on n vertices must contain at least kn + k(k − 2) arcs, and we give a family of k-freezable oriented graphs that reach this bound. In the general case, we prove as a partial result that every non 2-mixing oriented graph has maximum average degree at least 7 2 .
Let D = (V, A) be a digraph. We define ∆max(D) as the maximum of {max(dIt is known that the dichromatic number of D is at most ∆min(D) + 1. In this work, we prove that every digraph D which has dichromatic number exactly ∆min(D) + 1 must contain the directed join of ← → Kr and ← → Ks for some r, s such that r + s = ∆min(D) + 1. In particular, every oriented graph G with ∆min( G) ≥ 2 has dichromatic number at most ∆min( G).Let G be an oriented graph of order n such that ∆min( G) ≤ 1. Given two 2-dicolourings of 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 G on n vertices, the distance between two k-dicolourings is at most 2∆min( G)n when k ≥ ∆min( G) + 1.We then extend a theorem of Feghali to digraphs. We prove that, for every digraph D with ∆max(D) = ∆ ≥ 3 and every k ≥ ∆ + 1, the k-dicolouring graph of D consists of isolated vertices and at most one further component that has diameter at most c∆n 2 , where c∆ = O(∆ 2 ) is a constant depending only on ∆.
Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$ and with $m(G)$ edges. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(d+\frac{\min\left\{ 2-d-2\sqrt{1-d},\sqrt{d}-d\right\}}{2\Delta+1}-O\left(\frac{1}{n}\right)\right)m(G)$, that is, this number visibly exceeds its expected value $d\cdot m(G)$ when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $\Delta\leq 2$, we present more detailed results.
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 $\Delta$.
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