Digraph redicolouring

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 >= delta*(min)(D)+2, generalizing a result due to Dyer et al. We also prove that every oriented graph (G) over right arrow is k-mixing for all k >= delta*(max)((G) over right arrow)+1 and for all k > delta(avg)*((G) over right arrow)+1. Here delta(min)*, delta(max)*, and delta(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 > delta(min)*(D) + 2 colours has diameter at most O(vertical bar V(D)vertical bar(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 >= 2 delta(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(delta (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 maxi-mum 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. (c) 2023 Elsevier Ltd. All rights reserved.