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ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 26(2) (2006) 281-289


Jozef Bucko

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

Peter Mihók

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

Mathematical Institute, Slovak Academy of Science
Gresákova 6, 040 01 Košice, Slovak Republic


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.


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Received 31 January 2005
Revised 2 December 2005