Uniquely $D$-Colourable Digraphs with Large Girth II: Simplification via Generalization

Abstract

We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $\varphi\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $\varphi=f\circ\psi$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with large girth, Canad. J. Math., 64(6) (2012), 1310—1328; MR2994666] analogously with how the work [J. Nešetřil and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161—172; MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with large girth, J. Graph Theory, 23(1) (1996), 33—41; MR1402136].

On Colouring Oriented Graphs of Large Girth

“…Subsequently to [1], a subset of the authors and their doctoral students in [5] completed work in the realm of digraphs analogous to that of Zhu for graphs in [13]. Then [6] generalized the results of [1,5] just as Nešetřil and Zhu in [9] generalized [4,13]. One of our successes in the present work is a similar sequence of generalizations for oriented graphs.…”

“…An attentive reader familiar with [6] may be concerned that our Theorem 1.1 is an immediate consequence of [6,Theorem 1]. After all, the earlier result applies to digraphs in general, and oriented graphs are a specific type of digraph.…”

“…The fact that we consider oriented colouring here means we have no need to turn to acyclic homomorphisms. The reader might appreciate how much this simplifies our proofs in comparison with those of [6].…”

“…In 1959, Paul Erdős [4] famously proved probabilistically the existence of graphs of arbitrarily large girth and arbitrarily large chromatic number. We briefly discuss the history of this and related topics and point the reader to [5] or [6] for more details and references. As [4] and many of its descendants give strictly nonconstructive proofs, one is led to seek constructions.…”

“…The former parameter is defined using 'acyclic' homomorphisms-see [1] for details-which introduced complications. For example, the authors of [6] had to use a lot of care to demonstrate that certain mappings don't fail to be acyclic homomorphisms. The fact that we consider oriented colouring here means we have no need to turn to acyclic homomorphisms.…”