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

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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


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.


[1] M. Blidia, M. Chellali and O. Favaron, Independence and 2-domination in trees, Australas. J. Combin. 33 (2005) 317-327.
[2] M. Blidia, M. Chellali and L. Volkmann, Some bounds on the p-domintion number in trees, Discrete Math. 306 (2006) 2031-2037, doi: 10.1016/j.disc.2006.04.010.
[3] R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Philos. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X.
[4] M. Chellali, Offensive alliances in bipartite graphs, J. Combin. Math. Combin. Comput., to appear.
[5] O. Favaron, G. Fricke, W. Goddard, S.M. Hedetniemi, S.T. Hedetniemi, P. Kristiansen, R.C. Laskar and D.R. Skaggs, Offensive alliances in graphs, Dicuss. Math. Graph Theory 24 (2004) 263-275, doi: 10.7151/dmgt.1230.
[6] O. Favaron, A. Hansberg and L. Volkmann, On k-domination and minimum degree in graphs, J. Graph Theory 57 (2008) 33-40, doi: 10.1002/jgt.20279.
[7] H. Fernau, J.A. Rodríguez and J.M. Sigarreta, Offensive r-alliance in graphs, Discrete Appl. Math. 157 (2009) 177-182, doi: 10.1016/j.dam.2008.06.001.
[8] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079.
[9] J. Fujisawa, A. Hansberg, T. Kubo, A. Saito, M. Sugita and L. Volkmann, Independence and 2-domination in bipartite graphs, Australas. J. Combin. 40 (2008) 265-268.
[10] A. Hansberg, D. Meierling and L. Volkmann, Independence and p-domination in graphs, submitted.
[11] P. Kristiansen, S. M. Hedetniemi and S. T. Hedetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004) 157-177.
[12] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).
[13] C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104.
[14] K. H. Shafique and R.D. Dutton, Maximum alliance-free and minimum alliance-cover sets, Congr. Numer. 162 (2003) 139-146.
[15] K. H. Shafique and R.D. Dutton, A tight bound on the cardinalities of maximum alliance-free and minimum alliance-cover sets, J. Combin. Math. Combin. Comput. 56 (2006) 139-145.

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