The total {k}-domatic number of a graph |
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Authors: | S M Sheikholeslami L Volkmann |
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Institution: | 1. Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, I.R. Iran 2. Lehrstuhl II f??r Mathematik, RWTH Aachen University, 52056, Aachen, Germany
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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 ?i=1dfi(v) £ k\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 dt{k}(G)d_{t}^{\{k\}}(G). Note that dt{1}(G)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 dt{k}(G)d_{t}^{\{k\}}(G). Many of the known bounds of d
t
(G) are immediate consequences of our results. |
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Keywords: | |
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