On infinite uniquely partitionable graphs and graph properties of finite character
Discussiones Mathematicae Graph Theory, Tome 29 (2009) no. 2, p. 241.

Voir la notice de l'article dans European Digital Mathematics Library

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 G [ V i ] ∈ i 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 i has a weakly universal graph.
Classification : 05C15, 05C75
Mots-clés : graph property of finite character, reducibility, uniquely partitionable graphs, weakly universal graph 
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Jozef Bucko; Peter Mihók. On infinite uniquely partitionable graphs and graph properties of finite character. Discussiones Mathematicae Graph Theory, Tome 29 (2009) no. 2, p. 241. https://geodesic-test.mathdoc.fr/item/DMGT_2009__29_2_270677/