Discussiones Mathematicae Graph Theory 29(2) (2009) 241-251
doi: 10.7151/dmgt.1444

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ON INFINITE UNIQUELY PARTITIONABLE GRAPHS AND GRAPH PROPERTIES OF FINITE CHARACTER

Jozef Bucko

Department of Applied Mathematics
Faculty of Economics, Technical University
B. Nemcovej, 040 01 Košice, Slovak Republic
e-mail: jozef.bucko@tuke.sk

Peter Mihók

Department of Applied Mathematics
Faculty of Economics, Technical University
B. Nemcovej, 040 01 Košice, Slovak Republic
and
Mathematical Institute, Slovak Academy of Science
Gresákova 6, 040 01 Košice, Slovak Republic
e-mail: peter.mihok@tuke.sk

Abstract

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 ºP2 ººPn the class of all (P1,P2,...,Pn)-partitionable graphs. A property ℜ = P1 ºP2 ººPn, 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.

Keywords: graph property of finite character, reducibility, uniquely partitionable graphs, weakly universal graph.

2000 Mathematics Subject Classification: 05C15, 05C75.

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Received 31 December 2007
Revised 27 November 2008
Accepted 1 December 2008