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Uniquely D-colourable Digraphs with Large Girth

Published online by Cambridge University Press:  20 November 2018

Ararat Harutyunyan ,
P. Mark Kayll ,
Bojan Mohar  and
Liam Rafferty
Ararat Harutyunyan
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6 email: aha43@sfu.camohar@sfu.ca
P. Mark Kayll
Affiliation:
Department of Mathematical Sciences, University of Montana, Missoula MT 59812-0864, USA email: mark.kayll@umontana.edurafferty@member.ams.org
Bojan Mohar
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6 email: aha43@sfu.camohar@sfu.ca
Liam Rafferty
Affiliation:
Department of Mathematical Sciences, University of Montana, Missoula MT 59812-0864, USA email: mark.kayll@umontana.edurafferty@member.ams.org
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Abstract

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Let $C$ and $D$ be digraphs. A mapping $f:V\left( D \right)\to V\left( C \right)$ is a $C$-colouring if for every arc $uv$ of $D$, either $f\left( u \right)f\left( v \right)$ is an arc of $C$ or $f\left( u \right)=f\left( v \right)$, 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 $r\ge 1$, there are uniquely circularly $r$-colourable digraphs with arbitrarily large girth.

Keywords

05C1505C2060C05digraph colouringacyclic homomorphismcircular chromatic numbergirth
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 6 , 01 December 2012 , pp. 1310 - 1328
Copyright
Copyright © Canadian Mathematical Society 2012

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