DMGT

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory  17(1) (1997)   103-113
DOI: https://doi.org/10.7151/dmgt.1043

UNIQUELY PARTITIONABLE GRAPHS

Jozef Bucko

Department of Geometry and Algebra, P.J. Šafárik University
Jesenná 5, 041 54 Košice, Slovak Republic
e-mail: [email protected]

Marietjie Frick

Department of Mathematics, Applied Mathematics and Astronomy
University of South Africa, P.O. Box 392, Pretoria, 0001 South Africa
e-mail: [email protected]

Peter Mihók and Roman Vasky

Department of Geometry and Algebra, P.J. Šafárik University
Jesenná 5, 041 54 Košice, Slovak Republic
e-mail: mihok@Košice.upjs.sk
e-mail: [email protected]

Abstract

Let P1,…,Pn be properties of graphs. A (P1,…,Pn)-partition of a graph G is a partition of the vertex set V(G) into subsets V1, …,Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1,…,n. A graph G is said to be uniquely (P1, …,Pn)-partitionable if G has exactly one (P1,…,Pn)-partition. A property P is called hereditary if every subgraph of every graph with property P also has property P. If every graph that is a disjoint union of two graphs that have property P also has property P, then we say that P is additive. A property P is called degenerate if there exists a bipartite graph that does not have property P. In this paper, we prove that if P1,…, Pn are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (P1,…,Pn)-partitionable graph.

Keywords: hereditary property of graphs, additivity, reducibility, vertex partition.

1991 Mathematics Subject Classification: 05C15, 05C70.

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