Sparse H-Colourable Graphs of Bounded Maximum Degree

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 5k13 such that G admits a surjective homomorphism c to F , and moreover, for any F -pointed graph H 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 ◦ c. As a consequence, we prove that if H is a projective graph of order k, then for any finite family F of prescribed mappings from a set X to V (H) (with |F| = t), there is a graph G of arbitrary large girth and of maximum degree at most 5k26mt (where m = |X|) such that X ⊆ 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 ∈ F .