Abstract.
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 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 of prescribed mappings from a set X to V(H) (with ||=t), there is a graph G of arbitrary large girth and of maximum degree at most 5k 26mt (where m=|X|) such that 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 .
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Supported in part by the National Science Council under grant NSC89-2115-M-110-012
Final version received: June 9, 2003
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Hajiabolhassan, H., Zhu, X. Sparse H-Colourable Graphs of Bounded Maximum Degree. Graphs and Combinatorics 20, 65–71 (2004). https://doi.org/10.1007/s00373-003-0542-z
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DOI: https://doi.org/10.1007/s00373-003-0542-z