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## Discussiones Mathematicae Graph Theory

2012 | 32 | 1 | 129-140
Tytuł artykułu

### The k-rainbow domatic number of a graph

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 ${f₁,f₂, ...,f_d}$ of k-rainbow dominating functions on G with the property that $∑_{i = 1}^d |f_i(v)| ≤ k$ 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 $d_{rk}(G)$. Note that $d_{r1}(G)$ 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 $d_{rk}(G)$. Many of the known bounds of d(G) are immediate consequences of our results.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
129-140
Opis fizyczny
Daty
wydano
2012
otrzymano
2010-09-10
poprawiono
2011-03-10
zaakceptowano
2011-03-15
Twórcy
• Department of Mathematics, Azarbaijan Univercity of Tarbiat, Moallem, Tarbriz, I.R. Iran
autor
• Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Bibliografia
• [1] C. Berge, Theory of Graphs and its Applications (Methuen, London, 1962).
• [2] B. Brešar, M.A. Henning and D.F. Rall, Rainbow domination in graphs, Taiwanese J. Math. 12 (2008) 213-225.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).
• [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).
• [11] D. B. West, Introduction to Graph Theory (Prentice-Hall, Inc, 2000).
• [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.
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Bibliografia
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