TY - JOUR

T1 - Uniquely D-colourable digraphs with large girth

AU - Harutyunyan, Ararat

AU - Kayll, P. Mark

AU - Mohar, Bojan

AU - Rafferty, Liam

PY - 2012/12

Y1 - 2012/12

N2 - Let C and D be digraphs. A mapping f : V(D) → 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 ≥ 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.

AB - Let C and D be digraphs. A mapping f : V(D) → 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 ≥ 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.

KW - Acyclic homomorphism

KW - Circular chromatic number

KW - Digraph colouring

KW - Girth

UR - http://www.scopus.com/inward/record.url?scp=84894586729&partnerID=8YFLogxK

U2 - 10.4153/CJM-2011-084-9

DO - 10.4153/CJM-2011-084-9

M3 - Article

AN - SCOPUS:84894586729

SN - 0008-414X

VL - 64

SP - 1310

EP - 1328

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

IS - 6

ER -