Discussiones Mathematicae Graph Theory 29(3) (2009) 597-613
doi: 10.7151/dmgt.1467

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BOUNDS ON THE GLOBAL OFFENSIVE k-ALLIANCE NUMBER IN GRAPHS

1 Mustapha Chellali,  2Teresa W. Haynes, 3 Bert Randerath  and  3Lutz Volkmann

1LAMDA-RO Laboratory, Department of Mathematics
University of Blida
B.P. 270, Blida, Algeria
e-mail: m_chellali@yahoo.com

2Department of Mathematics
East Tennessee State University
Johnson City, TN 37614 USA
e-mail: haynes@etsu.edu

3Lehrstuhl II für Mathematik
RWTH Aachen University
Templergraben 55, D-52056 Aachen, Germany
e-mail: randerath@informatik.uni-koeln.de
e-mail: volkm@math2.rwth-aachen.de

Abstract

Let G = (V(G),E(G)) be a graph, and let k ≥1 be an integer. A set S ⊆V(G) is called a global offensive k-alliance if |N(v)∩S| ≥ |N(v)-S|+k for every v∈V(G)-S, where N(v) is the neighborhood of v. The global offensive k-alliance number γok(G) is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on γok(G) in terms of order, maximum degree, independence number, chromatic number and minimum degree.

Keywords: global offensive k-alliance number, independence number, chromatic number.

2000 Mathematics Subject Classification: 05C69.

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Received 8 July 2008
Revised 8 December 2008
Accepted 8 December 2008