The k-rainbow domatic number of a graph
Seyyed Mahmoud Sheikholeslami ; Lutz Volkmann
Discussiones Mathematicae Graph Theory, Tome 32 (2012), p. 129-140 / Harvested from The Polish Digital Mathematics Library
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For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set 1,2, ...,k such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃u ∈ N(v)f(u) = 1,2, ...,k is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set f₁,f₂,...,fd of k-rainbow dominating functions on G with the property that ∑i=1d|fi(v)|≤k for each v ∈ V(G), is called a k-rainbow dominating family (of functions) on G. The maximum number of functions in a k-rainbow dominating family on G is the k-rainbow domatic number of G, denoted by drk(G). Note that dr1(G) is the classical domatic number d(G). In this paper we initiate the study of the k-rainbow domatic number in graphs and we present some bounds for drk(G). Many of the known bounds of d(G) are immediate consequences of our results.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:271010 
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Seyyed Mahmoud Sheikholeslami; Lutz Volkmann. The k-rainbow domatic number of a graph. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 129-140. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1591/

[000] [1] C. Berge, Theory of Graphs and its Applications (Methuen, London, 1962).

[001] [2] B. Brešar, M.A. Henning and D.F. Rall, Rainbow domination in graphs, Taiwanese J. Math. 12 (2008) 213-225. | Zbl 1163.05046

[002] [3] B. Brešar and T.K. Šumenjak, On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 2394-2400, doi: 10.1016/j.dam.2007.07.018. | Zbl 1126.05091

[003] [4] G.J. Chang, J. Wu and X. Zhu, Rainbow domination on trees, Discrete Appl. Math. 158 (2010) 8-12, doi: 10.1016/j.dam.2009.08.010. | Zbl 1226.05191

[004] [5] T. Chunling, L. Xiaohui, Y. Yuansheng and L. Meiqin, 2-rainbow domination of generalized Petersen graphs P(n,2), Discrete Appl. Math 157 (2009) 1932-1937, doi: 10.1016/j.dam.2009.01.020. | Zbl 1183.05061

[005] [6] E.J. Cockayne, P.J.P. Grobler, W.R. Gründlingh, J. Munganga and J.H. van Vuuren, Protection of a graph, Util. Math. 67 (2005) 19-32. | Zbl 1081.05083

[006] [7] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247-261, doi: 10.1002/net.3230070305. | Zbl 0384.05051

[007] [8] B. Hartnell and D.F. Rall, On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24 (2004) 389-402, doi: 10.7151/dmgt.1238. | Zbl 1063.05107

[008] [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). | Zbl 0890.05002

[009] [10] H.B. Walikar, B.D. Acharya and E. Sampathkumar, Recent Developments in the Theory of Domination in Graphs (in: MRI Lecture Notes in Math., Mahta Research Instit., Allahabad, 1979).

[010] [11] D. B. West, Introduction to Graph Theory (Prentice-Hall, Inc, 2000).

[011] [12] G. Xu, 2-rainbow domination of generalized Petersen graphs P(n,3), Discrete Appl. Math. 157 (2009) 2570-2573, doi: 10.1016/j.dam.2009.03.016. | Zbl 1209.05175