Uniquely D-colourable digraphs with large girth

TL;DR

This paper explores the properties of uniquely D-colourable digraphs with large girth, demonstrating the existence of such graphs for various colourability conditions and establishing connections to circular colourings.

Contribution

It proves the existence of large girth digraphs with specific colourability properties, including unique colourability and circular colourings, expanding understanding of digraph colourability.

Findings

Existence of arbitrarily large girth digraphs with certain colourability properties.

Uniqueness of colourings is preserved in large girth digraphs.

Existence of uniquely circularly r-colourable digraphs for all r ≥ 1.

Abstract

Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring if for every arc $uv$ of D, either $f(u)f(v)$ is an arc of C or $f(u)=f(v)$, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.