Some remarks on universal graphs

LetΓ be a class of countable graphs, and let ℱ(Γ) denote the class of all countable graphs that do not contain any subgraph isomorphic to a member ofΓ. Furthermore, letTΓ andHΓ denote the class of all subdivisions of graphs inΓ and the class of all graphs contracting to a member ofΓ, respectively. As the main result of this paper it is decided which of the classes ℱ(TKn) and ℱ(HKn),n≦ℵ0, contain a universal element. In fact, for ℱ(TK4)=ℱ(HK4) a strongly universal graph is constructed, whereas for 5≦n≦ℵ0 the classes ℱ(TKn) and ℱ(HKn) have no universal elements.