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2012 | 32 | 3 | 461-471
Tytuł artykułu

The total {k}-domatic number of digraphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For a positive integer k, a total {k}-dominating function of a digraph D is a function f from the vertex set V(D) to the set {0,1,2, ...,k} such that for any vertex v ∈ V(D), the condition $∑_{u ∈ N^{ -}(v)}f(u) ≥ k$ is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set ${f₁,f₂, ...,f_d}$ of total {k}-dominating functions of D with the property that $∑_{i = 1}^d f_i(v) ≤ k$ for each v ∈ V(D), is called a total {k}-dominating family (of functions) on D. The maximum number of functions in a total {k}-dominating family on D is the total {k}-domatic number of D, denoted by $dₜ^{{k}}(D)$. Note that $dₜ^{{1}}(D)$ is the classic total domatic number $dₜ(D)$. In this paper we initiate the study of the total {k}-domatic number in digraphs, and we present some bounds for $dₜ^{{k}}(D)$. Some of our results are extensions of well-know properties of the total domatic number of digraphs and the total {k}-domatic number of graphs.
Wydawca
Rocznik
Tom
32
Numer
3
Strony
461-471
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-03-31
poprawiono
2011-08-29
zaakceptowano
2011-08-30
Twórcy
  • Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tarbriz, I.R. Iran
  • Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Bibliografia
  • [1] H. Aram, S.M. Sheikholeslami and L. Volkmann, On the total {k}-domination and {k}-domatic number of a graph, Bull. Malays. Math. Sci. Soc. (to appear).
  • [2] J. Chen, X. Hou and N. Li, The total {k}-domatic number of wheels and complete graphs, J. Comb. Optim. (to appear).
  • [3] E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219, doi: 10.1002/net.3230100304.
  • [4] E.J. Cockayne, T.W. Haynes, S.T. Hedetniemi, Z. Shanchao and B. Xu, Extremal graphs for inequalities involving domination parameters, Discrete Math. 216 (2000) 1-10, doi: 10.1016/S0012-365X(99)00251-4.
  • [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in graphs (New York: Marcel Dekker, Inc., 1998).
  • [6] K. Jacob and S. Arumugam, Domatic number of a digraph, Bull. Kerala Math. Assoc. 2 (2005) 93-103.
  • [7] N. Li and X. Hou, On the total {k}-domination number of Cartesian products of graphs, J. Comb. Optim. 18 (2009) 173-178, doi: 10.1007/s10878-008-9144-2.
  • [8] S.M. Sheikholeslami and L. Volkmann, The total {k}-domatic number of a graph, J. Comb. Optim. 23 (2012) 252-260, doi: 10.1007/s10878-010-9352-4.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1618
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