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Let $F$ be a graph of order at most $k$. We prove that for any
integer $g$ there is a graph $G$ of girth at least $g$ and of
maximum degree at most $5k^{13}$ such that $G$ admits a surjective
homomorphism $c$ to $F$, and moreover, for any $F$-pointed graph
$H$ (see definition below) with at most $k$ vertices, and for any
homomorphism $h$ from $G$ to $H$ there is a unique homomorphism
$f$ from $F$ to $H$ such that $h=f \circ c$. As a consequence, we
prove that if $H$ is a projective graph of order $k$, then for any
finite family $\cal F$ of prescribed mappings from a set $X$ to
$V(H)$ (with $|\cal F|$$= t$), there is a graph $G$ of arbitrary
large girth and of maximum degree at most $5k^{26mt}$ (where $m
=|X|$) such that $X \subseteq V(G)$ and up to an automorphism of
$H$, there are exactly $t$ homomorphisms from $G$ to $H$, each of
which is an extension of an $f \in \cal F$.
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