On infinite uniquely partitionable graphs and graph properties of finite character

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property P is of finite character if a graph G has a property P if and only if every finite induced subgraph of G has a property P . Let P1,P2, . . . ,Pn be graph properties of finite character, a graph G is said to be (uniquely) (P1,P2, . . . ,Pn)partitionable if there is an (exactly one) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. Let us denote by R = P1◦P2◦ · · · ◦Pn the class of all (P1,P2, . . . ,Pn)-partitionable graphs. A property R = P1◦P2◦ · · · ◦Pn, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P1,P2, . . . ,Pn)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property R of finite character there exists a weakly universal countable graph if and only if each property Pi has a weakly universal graph. 242 J. Bucko and P. Mihók