Uniquely partitionable graphs

Bucko, Jozef; Frick, Marietjie; Mihók, Peter; Vasky, Roman

doi:10.7151/dmgt.1043

Cited by 8 publications

(5 citation statements)

References 23 publications

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“…The following proposition summarises the basic properties of infinite uniquely partitionable graphs. We omit here the simple proofs, which are the same as the proofs for finite uniquely colourable graphs (see [10] and [4,12]). Proposition 1.…”

Section: Preliminary Resultsmentioning

confidence: 99%

On infinite uniquely partitionable graphs and graph properties of finite character

Bucko

Mihók

2009

Discuss. Math. Graph Theory

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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 P 1 , P 2 ,. .. , P n be graph properties of finite character, a graph G is said to be (uniquely) (P 1 , P 2 ,. .. , P n)partitionable if there is an (exactly one) partition {V 1 , V 2 ,. .. , V n } of V (G) such that G[V i ] ∈ P i for i = 1, 2,. .. , n. Let us denote by R = P 1 •P 2 • • • • •P n the class of all (P 1 , P 2 ,. .. , P n)-partitionable graphs. A property R = P 1 •P 2 • • • • •P n , n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P 1 , P 2 ,. .. , P 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 P i has a weakly universal graph.

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Section: Preliminary Resultsmentioning

confidence: 99%

On infinite uniquely partitionable graphs and graph properties of finite character

Bucko

Mihók

2009

Discuss. Math. Graph Theory

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“…The following Theorem describes the basic properties of the uniquely partitionable graphs (see [3,4,5]).…”

Section: Preliminary Resultsmentioning

confidence: 99%

Criteria of the existence of uniquely partionable graphs with respect to additive induced-hereditary properties

Broere¹,

Bucko

Mihók

2002

Discuss. Math. Graph Theory

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Let P 1 , P 2 , . . . , P n be graph properties, a graph G is said to be uniquely (P 1 , P 2 , . . . , P n )-partitionable if there is exactly oneWe prove that for additive and induced-hereditary properties uniquely (P 1 , P 2 , . . . , P n )-partitionable graphs exist if and 32 I. Broere, J. Bucko and P. Mihók only if P i and P j are either coprime or equal irreducible properties of graphs for every i = j, i, j ∈ {1, 2, . . . , n}.

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“…This forcing is done as follows. Mihók proved in [2] that for any property P > O not divisible by O, there exists a graph uniquely partitionable into O • P. Take a copy of such a graph and make one vertex from the O part of it adjacent…”

Section: The Np-hardness Resultsmentioning

confidence: 99%