Homomorphism bounded classes of graphs

A class C of graphs is said to be H -bounded if each graph in the class C admits a homomorphism to H . We give a general necessary and sufficient condition for the existence of bounds with special local properties. This gives a new proof of the Häggkvist–Hell theorem [R. Häggkvist, P. Hell, Universality of A-mote graphs, European J. Combin. 14 (1993) 23–27] and implies several cases of the existence of triangle free bounds for planar graphs.