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## Discussiones Mathematicae Graph Theory

2009 | 29 | 2 | 241-251
Tytuł artykułu

### On infinite uniquely partitionable graphs and graph properties of finite character

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 𝓟₁,𝓟₂,...,𝓟ₙ be graph properties of finite character, a graph G is said to be (uniquely) (𝓟₁, 𝓟₂, ...,𝓟ₙ)-partitionable if there is an (exactly one) partition {V₁, V₂, ..., Vₙ} of V(G) such that $G[V_i] ∈ 𝓟_i$ for i = 1,2,...,n. Let us denote by ℜ = 𝓟₁ ∘ 𝓟₂ ∘ ... ∘ 𝓟ₙ the class of all (𝓟₁,𝓟₂,...,𝓟ₙ)-partitionable graphs. A property ℜ = 𝓟₁ ∘ 𝓟₂ ∘ ... ∘ 𝓟ₙ, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property ℜ of finite character has a uniquely (𝓟₁, 𝓟₂, ...,𝓟ₙ)-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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
241-251
Opis fizyczny
Daty
wydano
2009
otrzymano
2007-12-31
poprawiono
2008-11-27
zaakceptowano
2008-12-01
Twórcy
autor
• Department of Applied Mathematics, Faculty of Economics, Technical University, B. Nĕmcovej, 040 01 Košice, Slovak Republic
autor
• Department of Applied Mathematics, Faculty of Economics, Technical University, B. Nĕmcovej, 040 01 Košice, Slovak Republic
• Mathematical Institute, Slovak Academy of Science, Gresákova 6, 040 01 Košice, Slovak Republic
Bibliografia
• [1] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, Survey of hereditary graph properties, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.
• [2] I. Broere and J. Bucko, Divisibility in additive hereditary graph properties and uniquely partitionable graphs, Tatra Mt. Math. Publ. 18 (1999) 79-87.
• [3] I. Broere, J. Bucko and P. Mihók, Criteria for the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties, Discuss. Math. Graph Theory 22 (2002) 31-37, doi: 10.7151/dmgt.1156.
• [4] J. Bucko, M. Frick, P. Mihók and R. Vasky, Uniquely partitionable graphs, Discuss. Math. Graph Theory 17 (1997) 103-114, doi: 10.7151/dmgt.1043.
• [5] G. Cherlin, S. Shelah and N. Shi, Universal graphs with forbidden subgraphs and algebraic closure, Advances in Appl. Math. 22 (1999) 454-491, doi: 10.1006/aama.1998.0641.
• [6] R. Cowen, S.H. Hechler and P. Mihók, Graph coloring compactness theorems equivalent to BPI, Scientia Math. Japonicae 56 (2002) 171-180.
• [7] A. Farrugia, P. Mihók, R.B. Richter and G. Semanišin, Factorizations and characterizations of induced-hereditary and compositive properties, J. Graph Theory 49 (2005) 11-27, doi: 10.1002/jgt.20062.
• [8] B. Ganter and R. Wille, Formal Concept Analysis - Mathematical Foundation (Springer-Verlag Berlin Heidelberg, 1999), doi: 10.1007/978-3-642-59830-2.
• [9] R.L. Graham, M. Grotschel and L. Lovasz, Handbook of Combinatorics (Elsevier Science B.V., Amsterdam, 1995).
• [10] F. Harary, S.T. Hedetniemi and R.W. Robinson, Uniquely colourable graphs, J. Combin. Theory 6 (1969) 264-270, doi: 10.1016/S0021-9800(69)80086-4.
• [11] W. Imrich, P. Mihók and G. Semanišin, A note on the unique factorization theorem for properties of infinite graphs, Stud. Univ. Zilina, Math. Ser. 16 (2003) 51-54.
• [12] P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, hypergraphs and matroids (Zielona Góra, 1985) 49-58.
• [13] P. Mihók, Unique factorization theorem, Discuss. Math. Graph Theory 20 (2000) 143-154, doi: 10.7151/dmgt.1114.
• [14] P. Mihók, On the lattice of additive hereditary properties of object systems, Tatra Mt. Math. Publ. 30 (2005) 155-161.
• [15] P. Mihók and G. Semanišin, Unique Factorization Theorem and Formal Concept Analysis, CLA 2006, Yasmin, Hammamet, Tunisia, (2006), 195-203.
• [16] N.W. Sauer, Canonical vertex partitions, Combinatorics, Probability and Computing 12 (2003) 671-704, doi: 10.1017/S0963548303005765.
• [17] E.R. Scheinerman, On the structure of hereditary classes of graphs, J. Graph Theory 10 (1986) 545-551, doi: 10.1002/jgt.3190100414.
Typ dokumentu
Bibliografia
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