On universal graphs for hom-properties

A graph property is any isomorphism closed class of simple graphs. For a simple finite graph H , let → H denote the class of all simple countable graphs that admit homomorphisms to H , such classes of graphs are called hom-properties. Given a graph property P , a graph This research was supported in part by Slovak VEGA grant 2/7141/27. The research of the author was supported in part by VEGA Grant 1/0035/09, Slovak APVV grant 0007-07. 402 P. Mihók, J. Miškuf and G. Semanišin G ∈ P is universal in P if each member of P is isomorphic to an induced subgraph of G. In particular, we consider universal graphs in → H and we give a new proof of the existence of a universal graph in → H , for any finite graph H .