# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## The Total {k}-domatic Number of Digraphs

 Seyed Mahmoud Sheikholeslami Department of Mathematics Azarbaijan University of Tarbiat Moallem Tarbriz, I.R. Iran Lutz Volkmann Lehrstuhl II für Mathematik RWTH Aachen University 52056 Aachen, Germany

## Abstract

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, …,fd} of total {k}-dominating functions of D with the property that ∑i = 1dfi(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.

Keywords: digraph, total {k}-dominating function, total {k}-domination number, total {k}-domatic number

2010 Mathematics Subject Classification: 05C69.

## References

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