On uniquely partitionable relational structures and object systems

Jozef Bucko; Peter Mihók

Discussiones Mathematicae Graph Theory, Tome 26 (2006), p. 281-289 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $V (A_{i})$ of each object $A_{i} \in E$ is a finite set with at least two elements and $V \supseteq ⋃_{i = 1}^{m} V (A_{i})$ . 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.

Publié le : 2006-01-01

Zbl 1142.05024

EUDML-ID : urn:eudml:doc:270588

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Jozef Bucko; Peter Mihók. On uniquely partitionable relational structures and object systems. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 281-289. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1320/

Bibliographie

[000] [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. | Zbl 0885.05062

[001] [2] B. Bollobás and A. G. Thomason, Uniquely partitionable graphs, J. London Math. Soc. (2) 16 (1977) 403-410. | Zbl 0377.05038

[002] [3] A. Bonato, Homomorphism and amalgamation, Discrete Math. 270 (2003) 33-42, doi: 10.1016/S0012-365X(02)00864-6.

[003] [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. | Zbl 0902.05026

[004] [5] J. I. Brown and D. G. Corneil, On generalized graph colourings, J. Graph Theory 11 (1987) 86-99, doi: 10.1002/jgt.3190110113.

[005] [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. | Zbl 1013.05066

[006] [7] A. Farrugia, Uniqueness and complexity in generalised colouring, Ph.D. thesis, University of Waterloo, April 2003 (available at http://etheses.uwaterloo.ca).

[007] [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. | Zbl 1067.05057

[008] [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. | Zbl 1061.05070

[009] [10] R. Fraïssé, Sur certains relations qui generalisent l'ordre des nombers rationnels, C.R. Acad. Sci. Paris 237 (1953) 540-542. | Zbl 0051.03803

[010] [11] R. Fraïssé, Theory of Relations (North-Holland, Amsterdam, 1986).

[011] [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. | Zbl 0175.50205

[012] [13] J. Jakubík, On the lattice of additive hereditary properties of finite graphs, Discuss. Math. General Algebra and Applications 22 (2002) 73-86. | Zbl 1032.06003

[013] [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. | Zbl 0949.05025

[014] [15] P. Mihók, On the lattice of additive hereditary properties of object-systems, Tatra Mountains Math. Publ. 30 (2005) 155-161. | Zbl 1150.05394

[015] [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. | Zbl 0623.05043

[016] [17] P. Mihók, Reducible properties and uniquely partitionable graphs, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 49 (1999) 213-218. | Zbl 0937.05042

[017] [18] P. Mihók Unique factorization theorems, Discuss. Math. Graph Theory 20 (2000) 143-153, doi: 10.7151/dmgt.1114.

[018] [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 | Zbl 0942.05056

[019] [20] J. Mitchem, Uniquely k-arborable graphs, Israel J. Math. 10 (1971) 17-25, doi: 10.1007/BF02771516. | Zbl 0224.05105

[020] [21] B.C. Pierce, Basic Category Theory for Computer Scientists (Foundations of Computing Series, The MIT Press, Cambridge, Massachusetts 1991).

[021] [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.

[022] [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. | Zbl 1054.05047