DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 26(2) (2006) 281-289
DOI: 10.7151/dmgt.1320

ON UNIQUELY PARTITIONABLE RELATIONAL STRUCTURES AND OBJECT SYSTEMS

Jozef Bucko

Department of Applied Mathematics
Faculty of Economics, Technical University
B. Nemcovej, 040 01 Košice, Slovak Republic
e-mail: Jozef.Bucko@tuke.sk

Peter Mihók

Department of Applied Mathematics
Faculty of Economics, Technical University
B. Nemcovej, 040 01 Košice, Slovak Republic

and
Mathematical Institute, Slovak Academy of Science
Gresákova 6, 040 01 Košice, Slovak Republic
e-mail: Peter.Mihok@tuke.sk

Abstract

We introduce object systems as a common generalization of graphs, hypergraphs, digraphs and relational structures. Let C be a concrete category, a simple object system over C is an ordered pair S = (V,E), where E = {A1,A2,...,Am} is a finite set of the objects of C, such that the ground-set V(Ai) of each object Ai ∈ E is a finite set with at least two elements and V ⊇ ∪i = 1m V(Ai). To generalize the results on graph colourings to simple object systems we define, analogously as for graphs, that an additive induced-hereditary property of simple object systems over a category C is any class of systems closed under isomorphism, induced-subsystems and disjoint union of systems, respectively. We present a survey of recent results and conditions for object systems to be uniquely partitionable into subsystems of given properties.

Keywords: graph, digraph, hypergraph, vertex colouring, uniquely partitionable system.

2000 Mathematics Subject Classification: 05C15, 05C20, 05C65, 05C75.

References

[1] D. Achlioptas, J.I. Brown, D.G. Corneil and M.S.O. Molloy, The existence of uniquely − G colourable graphs, Discrete Math. 179 (1998) 1-11, doi: 10.1016/S0012-365X(97)00022-8.
[2] B. Bollobás and A. G. Thomason, Uniquely partitionable graphs, J. London Math. Soc. (2) 16 (1977) 403-410.
[3] A. Bonato, Homomorphism and amalgamation, Discrete Math. 270 (2003) 33-42, doi: 10.1016/S0012-365X(02)00864-6.
[4] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, Survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.
[5] J. I. Brown and D. G. Corneil, On generalized graph colourings, J. Graph Theory 11 (1987) 86-99, doi: 10.1002/jgt.3190110113.
[6] 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.
[7] A. Farrugia, Uniqueness and complexity in generalised colouring, Ph.D. thesis, University of Waterloo, April 2003 (available at http://etheses.uwaterloo.ca).
[8] A. Farrugia, P. Mihók, R.B. Richter and G. Semanišin, Factorisations and characterisations of induced-hereditary and compositive properties, J. Graph Theory 49 (2005) 11-27, doi: 10.1002/jgt.20062.
[9] A. Farrugia and R.B. Richter, Unique factorisation of additive induced-hereditary properties, Discuss. Math. Graph Theory 24 (2004) 319-343, doi: 10.7151/dmgt.1234.
[10] R. Fraïssé, Sur certains relations qui generalisent l'ordre des nombers rationnels, C.R. Acad. Sci. Paris 237 (1953) 540-542.
[11] R. Fraïssé, Theory of Relations (North-Holland, Amsterdam, 1986).
[12] 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.
[13] J. Jakubík, On the lattice of additive hereditary properties of finite graphs, Discuss. Math. General Algebra and Applications 22 (2002) 73-86.
[14] J. Kratochvíl and P. Mihók, Hom-properties are uniquely factorizable into irreducible factors, Discrete Math. 213 (2000) 189-194, doi: 10.1016/S0012-365X(99)00179-X.
[15] P. Mihók, On the lattice of additive hereditary properties of object-systems, Tatra Mountains Math. Publ. 30 (2005) 155-161.
[16] 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.
[17] P. Mihók, Reducible properties and uniquely partitionable graphs, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 49 (1999) 213-218.
[18] P. Mihók Unique factorization theorems, Discuss. Math. Graph Theory 20 (2000) 143-153, doi: 10.7151/dmgt.1114.
[19] P. Mihók, G. Semanišin and R. Vasky, Additive and Hereditary Properties of Graphs are Uniquely Factorizable into Irreducible Factors, J. Graph Theory 33 (2000) 44-53, doi: 10.1002/(SICI)1097-0118(200001)33:1<44::AID-JGT5>3.0.CO;2-O.
[20] J. Mitchem, Uniquely k-arborable graphs, Israel J. Math. 10 (1971) 17-25, doi: 10.1007/BF02771516.
[21] B.C. Pierce, Basic Category Theory for Computer Scientists (Foundations of Computing Series, The MIT Press, Cambridge, Massachusetts 1991).
[22] J.M.S. Simoes-Pereira, On graphs uniquely partitionable into n-degenerate subgraphs, in: Infinite and Finite Sets, Colloquia Math. Soc. J. Bólyai 10 (1975) 1351-1364.
[23] R. Vasky, Unique factorization theorem for additive induced-hereditary properties of digraphs Studies of the University of Zilina, Mathematical Series 15 (2002) 83-96.

Received 31 January 2005
Revised 2 December 2005