Homomorphism bounded classes of graphs

A class CC of graphs is said to be HH-bounded   if each graph in the class CC admits a homomorphism to HH. 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.