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Discussiones Mathematicae Graph Theory 26(2) (2006) 281-289
DOI: https://doi.org/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,...,A_m} is a finite set of the objects of C, such that the ground-set V(A_i) of each object A_i ∈ E is a finite set with at least two elements and V ⊇ ∪_{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.

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