Affiliation:
1. CNRS, I3S, INRIA Université Côte d'Azur Sophia Antipolis Nice France
Abstract
AbstractLet be a digraph. We define as the maximum of and as the maximum of . It is known that the dichromatic number of is at most . In this work, we prove that every digraph which has dichromatic number exactly must contain the directed join of and for some such that , except if in which case must contain a digon. In particular, every oriented graph with has dichromatic number at most . Let be an oriented graph of order such that . Given two 2‐dicolourings of , we show that we can transform one into the other in at most steps, by recolouring exactly one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph on vertices, the distance between two ‐dicolourings is at most when . We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph with and every , the ‐dicolouring graph of consists of isolated vertices and at most one further component that has diameter at most , where is a constant depending only on .
Subject
Geometry and Topology,Discrete Mathematics and Combinatorics
Reference17 articles.
1. Four proofs of the directed Brooks' theorem;Aboulker P.;Discrete Math,2022
2. V.Bartier Combinatorial and algorithmic aspects of reconfiguration Ph.D. thesis Université Grenoble Alpes (2021).
3. The circular chromatic number of a digraph
4. Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs
5. N.Bousquet L.Feuilloley M.Heinrich andM.Rabie Short and local transformations between (Δ+1${\rm{\Delta }}+1$)‐colorings arXiv preprint arXiv:2203.08885 (2022).