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    Wyświetlanie 1-7 z 7
    Tytuł:
    Universality in Graph Properties with Degree Restrictions
    Autorzy:
    Broere, Izak
    Heidema, Johannes
    Mihók, Peter
    Powiązania:
    https://bibliotekanauki.pl/articles/30146518.pdf
    Data publikacji:
    2013-07-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    countable graph
    universal graph
    induced-hereditary
    k-degenerate graph
    graph with colouring number at most k + 1
    graph property with assignment
    Opis:
    Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set ℐc of all countable graphs (since every graph in ℐc is isomorphic to an induced subgraph of R). A brief overview of known universality results for some induced-hereditary subsets of ℐc is provided. We then construct a k-degenerate graph which is universal for the induced-hereditary property of finite k-degenerate graphs. In order to attempt the corresponding problem for the property of countable graphs with colouring number at most k + 1, the notion of a property with assignment is introduced and studied. Using this notion, we are able to construct a universal graph in this graph property and investigate its attributes.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 3; 477-492
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Universality for and in Induced-Hereditary Graph Properties
    Autorzy:
    Broere, Izak
    Heidema, Johannes
    Powiązania:
    https://bibliotekanauki.pl/articles/30146860.pdf
    Data publikacji:
    2013-03-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    countable graph
    universal graph
    induced-hereditary property
    Opis:
    The well-known Rado graph $R$ is universal in the set of all countable graphs \( \mathcal{I} \), since every countable graph is an induced subgraph of $R$. We study universality in \( \mathcal{I} \) and, using $R$, show the existence of $2^{\aleph_0}$ pairwise non-isomorphic graphs which are universal in \( \mathcal{I} \) and denumerably many other universal graphs in \( \mathcal{I} \) with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are $ 2^{2^{\aleph_0 } }$ properties in the lattice $ \mathbb{K}_\le $ of induced-hereditary properties of which only at most $ 2^{\aleph_0} $ contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 1; 33-47
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Unique factorization theorem
    Autorzy:
    Mihók, Peter
    Powiązania:
    https://bibliotekanauki.pl/articles/743745.pdf
    Data publikacji:
    2000
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    induced-hereditary
    additive property of graphs
    reducible property of graphs
    unique factorization
    uniquely partitionable graphs
    generating sets
    Opis:
    A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G[V_i]$ of G induced by V_i belongs to $_i$; i = 1,2,...,n. A property is said to be reducible if there exist properties ₁ and ₂ such that = ₁ º₂; otherwise the property is irreducible. We prove that every additive and induced-hereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (₁,₂, ...,ₙ)-partitionable graphs for any irreducible properties ₁,₂, ...,ₙ.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2000, 20, 1; 143-154
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Quest for A Characterization of Hom-Properties of Finite Character
    Autorzy:
    Broere, Izak
    Matsoha, Moroli D.V.
    Heidema, Johannes
    Powiązania:
    https://bibliotekanauki.pl/articles/31340894.pdf
    Data publikacji:
    2016-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    (countable) graph
    homomorphism (of graphs)
    property of graphs
    hom-property
    (finitely-)induced-hereditary property
    finitely determined property
    (weakly) finite character
    axiomatizable property
    compactness theorems
    core
    connectedness
    chromatic number
    clique number
    independence number
    dominating set
    Opis:
    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 \( G \rightarrow H \). Given any graph \( H \), the hom-property \( \rightarrow H \) is the set of \( H \)-colourable graphs, i.e., the set of all graphs \( G \) satisfying \( G \rightarrow H \). A graph property \( mathcal{P} \) is of finite character if, whenever we have that \( F \in \mathcal{P} \) for every finite induced subgraph \( F \) of a graph \( G \), then we have that \( G \in \mathcal{P} \) 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 \( \rightarrow H \) 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 \( \rightarrow H \) 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.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2016, 36, 2; 479-500
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Prime ideals in the lattice of additive induced-hereditary graph properties
    Autorzy:
    Berger, Amelie
    Mihók, Peter
    Powiązania:
    https://bibliotekanauki.pl/articles/743387.pdf
    Data publikacji:
    2003
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    hereditary graph property
    prime ideal
    distributive lattice
    induced subgraphs
    Opis:
    An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded join-irreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2003, 23, 1; 117-127
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On generating sets of induced-hereditary properties
    Autorzy:
    Semanišin, Gabriel
    Powiązania:
    https://bibliotekanauki.pl/articles/743561.pdf
    Data publikacji:
    2002
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    induced-hereditary property of graphs
    additivity
    reducibility
    generating sets
    maximal graphs
    unique factorization
    Opis:
    A natural generalization of the fundamental graph vertex-colouring problem leads to the class of problems known as generalized or improper colourings. These problems can be very well described in the language of reducible (induced) hereditary properties of graphs. It turned out that a very useful tool for the unique determination of these properties are generating sets. In this paper we focus on the structure of specific generating sets which provide the base for the proof of The Unique Factorization Theorem for induced-hereditary properties of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2002, 22, 1; 183-192
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Gallais innequality for critical graphs of reducible hereditary properties
    Autorzy:
    Mihók, Peter
    Skrekovski, Riste
    Powiązania:
    https://bibliotekanauki.pl/articles/743466.pdf
    Data publikacji:
    2001
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    additive induced-hereditary property of graphs
    reducible property of graphs
    critical graph
    Gallai's Theorem
    Opis:
    In this paper Gallai's inequality on the number of edges in critical graphs is generalized for reducible additive induced-hereditary properties of graphs in the following way. Let $₁,₂,...,ₖ$ (k ≥ 2) be additive induced-hereditary properties, $ = ₁ ∘ ₂ ∘ ... ∘ₖ$ and $δ = ∑_{i=1}^k δ(_i)$. Suppose that G is an -critical graph with n vertices and m edges. Then 2m ≥ δn + (δ-2)/(δ²+2δ-2)*n + (2δ)/(δ²+2δ-2) unless = ² or $G = K_{δ+1}$. The generalization of Gallai's inequality for -choice critical graphs is also presented.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2001, 21, 2; 167-177
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-7 z 7

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