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].
Homomorphisms from sparse graphs with large girth
