On sparse graphs with given colorings and homomorphisms

We prove that for every graph H and positive integers k and l there exists a graph G with girth at least l such that for all graphs H′ with at most k vertices there exists a homomorphism G→H′ if and only if there exists a homomorphism H→H′. This implies (for H=Kk) the classical result of Erdős and other generalizations (such as Sparse Incomparability Lemma). We refine the above statement to the 1-1 correspondence between the set of all homomorphisms G→H′ and the set of all homomorphisms H→H′. This in turn yields the existence of sparse uniquely H-colorable graphs and, perhaps surprisingly, provides a characterization of the graphs H for which the analog of Müllerʹs theorem holds for H-colorings.