Digraph redicolouring

Abstract

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≥δmin∗(D)+2, generalizing a result due to Dyer et al. We also prove that every oriented graph G→ is k-mixing for all k≥δmax∗(G→)+1 and for all k≥δavg∗(G→)+1. Here δmin∗, δmax∗, and δavg∗ 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≥δmin∗(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≥2δmin∗(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≥32(δmin∗(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 72.