The total {k}-domatic number of a graph

S. M. Sheikholeslami () and L. Volkmann ()
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S. M. Sheikholeslami: Azarbaijan University of Tarbiat Moallem
L. Volkmann: RWTH Aachen University

Journal of Combinatorial Optimization, 2012, vol. 23, issue 2, No 6, 252-260

Abstract: Abstract For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f 1,f 2,…,f d } of total {k}-dominating functions on G with the property that $\sum_{i=1}^{d}f_{i}(v)\le k$ for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by $d_{t}^{\{k\}}(G)$ . Note that $d_{t}^{\{1\}}(G)$ is the classic total domatic number d t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for $d_{t}^{\{k\}}(G)$ . Many of the known bounds of d t (G) are immediate consequences of our results.

Keywords: Total {k}-dominating function; Total {k}-domination number; Total {k}-domatic number (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10878-010-9352-4

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