# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## 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.

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