A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property is of finite character if a graph G has a property if and only if every finite induced subgraph of G has a property . Let ₁,₂,...,ₙ be graph properties of finite character, a graph G is said to be (uniquely) (₁, ₂, ...,ₙ)-partitionable if there is an (exactly one) partition V₁, V₂, ..., Vₙ of V(G) such that for i = 1,2,...,n. Let us denote by ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ the class of all (₁,₂,...,ₙ)-partitionable graphs. A property ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property ℜ of finite character has a uniquely (₁, ₂, ...,ₙ)-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 has a weakly universal graph.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1444,
author = {Jozef Bucko and Peter Mih\'ok},
title = {On infinite uniquely partitionable graphs and graph properties of finite character},
journal = {Discussiones Mathematicae Graph Theory},
volume = {29},
year = {2009},
pages = {241-251},
zbl = {1194.05037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1444}
}
Jozef Bucko; Peter Mihók. On infinite uniquely partitionable graphs and graph properties of finite character. Discussiones Mathematicae Graph Theory, Tome 29 (2009) pp. 241-251. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1444/
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