Geodesic
On infinite uniquely partitionable graphs and graph properties of finite character
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 2, pp. 241-251.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: graph property of finite character, reducibility, uniquely partitionable graphs, weakly universal graph 
@article{DMGT_2009_29_2_a2,
     author = {Bucko, Jozef and Mih\'ok, Peter},
     title = {On infinite uniquely partitionable graphs and graph properties of finite character},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {241--251},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2009_29_2_a2/}
}
TY  - JOUR
AU  - Bucko, Jozef
AU  - Mihók, Peter
TI  - On infinite uniquely partitionable graphs and graph properties of finite character
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2009
SP  - 241
EP  - 251
VL  - 29
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2009_29_2_a2/
LA  - en
ID  - DMGT_2009_29_2_a2
ER  - 
%0 Journal Article
%A Bucko, Jozef
%A Mihók, Peter
%T On infinite uniquely partitionable graphs and graph properties of finite character
%J Discussiones Mathematicae. Graph Theory
%D 2009
%P 241-251
%V 29
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2009_29_2_a2/
%G en
%F DMGT_2009_29_2_a2
Bucko, Jozef; Mihók, Peter. On infinite uniquely partitionable graphs and graph properties of finite character. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 2, pp. 241-251. http://geodesic.mathdoc.fr/item/DMGT_2009_29_2_a2/

[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.