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. © 2005 Elsevier Ltd. All rights reserved.
Marshall, T., Nasraser, R., & Nešetřil, J. (2006). Homomorphism bounded classes of graphs. European Journal of Combinatorics, 27(4), 592–600. https://doi.org/10.1016/j.ejc.2004.07.014