PDF

Discussiones Mathematicae Graph Theory 26(1) (2006) 59-72
DOI: https://doi.org/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 = C_m ×K_n, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = C_m×K_n,c) with c > χ(G), where n ≥ 2 and m ≥ 3, unless the defining number d(K₃×C_2r,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

[1]	J. Cooper, D. Donovan and J. Seberry, Latin squares and critical sets of minimal size, Austral. J. Combin. 4 (1991) 113-120.
[2]	M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graph, Ars Combin. 51 (1999) 295-305.
[3]	M. Mahdian, E.S. Mahmoodian, R. Naserasr and F. Harary, On defining sets of vertex colorings of the cartesian product of a cycle with a complete graph, Combinatorics, Graph Theory and Algorithms (1999) 461-467.
[4]	E.S. Mahmoodian and E. Mendelsohn, On defining numbers of vertex coloring of regular graphs, 16th British Combinatorial Conference (London, 1997). Discrete Math. 197/198 (1999) 543-554.
[5]	E.S. Mahmoodian, R. Naserasr and M. Zaker, Defining sets in vertex colorings of graphs and Latin rectangles, Discrete Math. (to appear).
[6]	E. Mendelsohn and D.A. Mojdeh, On defining spectrum of regular graph, (submitted).
[7]	D.A. Mojdeh, On conjectures of the defining set of (vertex) graph colourings, Austral. J. Combin. (to appear).
[8]	A.P. Street, Defining sets for block designs; an update, in: C.J. Colbourn, E.S. Mahmoodian (eds), Combinatorics advances, Mathematics and its applications (Kluwer Academic Publishers, Dordrecht, 1995) 307-320.
[9]	D.B. West, Introduction to Graph Theory (Second Edition) (Prentice Hall, USA, 2001).

Received 6 November 2004
Revised 13 September 2005