# The total {k}-domatic number of digraphs

Seyed Mahmoud Sheikholeslami; Lutz Volkmann

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 3, page 461-471
- ISSN: 2083-5892

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topSeyed Mahmoud Sheikholeslami, and Lutz Volkmann. "The total {k}-domatic number of digraphs." Discussiones Mathematicae Graph Theory 32.3 (2012): 461-471. <http://eudml.org/doc/271065>.

@article{SeyedMahmoudSheikholeslami2012,

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 $\{f₁,f₂, ...,f_d\}$ 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.},

author = {Seyed Mahmoud Sheikholeslami, Lutz Volkmann},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {digraph; total \{k\}-dominating function; total \{k\}-domination number; total \{k\}-domatic number; total -dominating function; total -domination number; total -domatic number},

language = {eng},

number = {3},

pages = {461-471},

title = {The total \{k\}-domatic number of digraphs},

url = {http://eudml.org/doc/271065},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Seyed Mahmoud Sheikholeslami

AU - Lutz Volkmann

TI - The total {k}-domatic number of digraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 3

SP - 461

EP - 471

AB - 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.

LA - eng

KW - digraph; total {k}-dominating function; total {k}-domination number; total {k}-domatic number; total -dominating function; total -domination number; total -domatic number

UR - http://eudml.org/doc/271065

ER -

## References

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- [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. Zbl1243.90229

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