DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 26(1) (2006) 59-72
DOI: 10.7151/dmgt.1301

DEFINING SETS IN (PROPER) VERTEX COLORINGS OF THE CARTESIAN PRODUCT OF A CYCLE WITH A COMPLETE GRAPH

D. Ali Mojdeh

Department of Mathematics
University of Mazandaran
Babolsar, IRAN, P.O. Box 47416-1467
e-mail: dmojdeh@umz.ac.ir

Abstract

In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c).

The d(G = Cm ×Kn, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cm×Kn,c) with c > χ(G), where n ≥ 2 and m ≥ 3, unless the defining number d(K3×C2r,4), which is given an upper and lower bounds for this defining number. Also some bounds of defining number are introduced.

Keywords: graph coloring, defining set, cartesian product.

2000 Mathematics Subject Classification: 05C15, 05C38.

References

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Received 6 November 2004
Revised 13 September 2005