A note on classes of subgraphs of locally finite graphs

We investigate the question how `small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph $H$ containing all elements of a graph class $\mathcal G$. These conditions imply that such a graph $H$ exists for the class $\mathcal G_d$ consisting of all graphs with maximum degree $