On Universal Positive Graphs

We study the existence of the universal computable numberings and the universal graphs for various classes of positive graphs. It is known that each -axiomatizable class of graphs can be characterized as follows: A graph belongs to if and only if for a given family of finite graphs no graph in is isomorphically embeddable into .If all graphs in are weakly connected; then, under additional effectiveness conditions, the corresponding class has some universal computable numbering and universal positive graph. The effectiveness conditions hold for forests, bipartite graphs, planar graphs, and -colorable graphs (for a fixed number ). If is a finite family of the graphs with weakly connected complement then the corresponding class contains a universal positive graph (in general, a universal computable numbering for may fail to exist).