Uniquely D-colourable digraphs with large girthReport as inadecuate



 Uniquely D-colourable digraphs with large girth


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Let C and D be digraphs. A mapping $f:VD\to VC$ is a C-colouring if for every arc $uv$ of D, either $fufv$ is an arc of C or $fu=fv$, 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.



Author: Ararat Harutyunyan; P. Mark Kayll; Bojan Mohar; Liam Rafferty

Source: https://archive.org/



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