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1
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The decomposability of additive hereditary properties of graphs

100%
Broere I., Dorfling M.
Discussiones Mathematicae Graph Theory
|
2000
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tom 20
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nr 2
281-291
EN
An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If 𝓟₁,...,𝓟ₙ are properties of graphs, then a (𝓟₁,...,𝓟ₙ)-decomposition of a graph G is a partition E₁,...,Eₙ of E(G) such that $G[E_i]$, the subgraph of G induced by $E_i$, is in $𝓟_i$, for i = 1,...,n. We define 𝓟₁ ⊕...⊕ 𝓟ₙ as the property {G ∈ 𝓘: G has a (𝓟₁,...,𝓟ₙ)-decomposition}. A property 𝓟 is said to be decomposable if there exist non-trivial hereditary properties 𝓟₁ and 𝓟₂ such that 𝓟 = 𝓟₁⊕ 𝓟₂. We study the decomposability of the well-known properties of graphs 𝓘ₖ, 𝓞ₖ, 𝓦ₖ, 𝓣ₖ, 𝓢ₖ, 𝓓ₖ and $𝓞 ^{p}$.
2
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Generalized chromatic numbers and additive hereditary properties of graphs

100%
Broere I., Dorfling S., Jonck E.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 2
259-270
EN
An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let 𝓟 and 𝓠 be additive hereditary properties of graphs. The generalized chromatic number $χ_{𝓠}(𝓟)$ is defined as follows: $χ_{𝓠}(𝓟) = n$ iff 𝓟 ⊆ 𝓠 ⁿ but $𝓟 ⊊ 𝓠^{n-1}$. We investigate the generalized chromatic numbers of the well-known properties of graphs 𝓘ₖ, 𝓞ₖ, 𝓦ₖ, 𝓢ₖ and 𝓓ₖ.
3
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Generalized edge-chromatic numbers and additive hereditary properties of graphs

100%
Dorfling M., Dorfling S.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 2
349-359
EN
An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let 𝓟 and 𝓠 be hereditary properties of graphs. The generalized edge-chromatic number $ρ'_{𝓠}(𝓟)$ is defined as the least integer n such that 𝓟 ⊆ n𝓠. We investigate the generalized edge-chromatic numbers of the properties → H, 𝓘ₖ, 𝓞ₖ, 𝓦*ₖ, 𝓢ₖ and 𝓓ₖ.
4
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On a characterization of graphs by average labellings

100%
Harminc M.
Discussiones Mathematicae Graph Theory
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1997
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tom 17
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nr 1
133-136
EN
The additive hereditary property of linear forests is characterized by the existence of average labellings.
5
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Factorizations of properties of graphs

88%
Broere I., Teboho Moagi S., Mihók P., Vasky R.
Discussiones Mathematicae Graph Theory
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1999
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tom 19
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nr 2
167-174
EN
A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs 𝓟₁,𝓟₂,...,𝓟ₙ a vertex (𝓟₁, 𝓟₂, ...,𝓟ₙ)-partition of a graph G is a partition {V₁,V₂,...,Vₙ} of V(G) such that for each i = 1,2,...,n the induced subgraph $G[V_i]$ has property $𝓟_i$. The class of all graphs having a vertex (𝓟₁, 𝓟₂, ...,𝓟ₙ)-partition is denoted by 𝓟₁∘𝓟₂∘...∘𝓟ₙ. A property 𝓡 is said to be reducible with respect to a lattice of properties of graphs 𝕃 if there are n ≥ 2 properties 𝓟₁,𝓟₂,...,𝓟ₙ ∈ 𝕃 such that 𝓡 = 𝓟₁∘𝓟₂∘...∘𝓟ₙ; otherwise 𝓡 is irreducible in 𝕃. We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.
6
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Remarks on the existence of uniquely partitionable planar graphs

88%
Borowiecki M., Mihók P., Tuza Z., Voigt M.
Discussiones Mathematicae Graph Theory
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1999
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tom 19
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nr 2
159-166
EN
We consider the problem of the existence of uniquely partitionable planar graphs. We survey some recent results and we prove the nonexistence of uniquely (𝓓₁,𝓓₁)-partitionable planar graphs with respect to the property 𝓓₁ "to be a forest".
7
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The Quest for A Characterization of Hom-Properties of Finite Character

51%
Broere I., Matsoha M. D. V., Heidema J.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
479-500
EN
A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graph H is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F ∈ P for every finite induced subgraph F of a graph G, then we have that G ∈ P too. We explore some of the relationships of the property attribute of being of finite character to other property attributes such as being finitely-induced-hereditary, being finitely determined, and being axiomatizable. We study the hom-properties of finite character, and prove some necessary and some sufficient conditions on H for →H to be of finite character. A notable (but known) sufficient condition is that H is a finite graph, and our new model-theoretic proof of this compactness result extends from hom-properties to all axiomatizable properties. In our quest to find an intrinsic characterization of those H for which →H is of finite character, we find an example of an infinite connected graph with no finite core and chromatic number 3 but with hom-property not of finite character.
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