The total {k}-domatic number of digraphs

Seyed 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 -