# The Signed Total Roman k-Domatic Number Of A Graph

• Volume: 37, Issue: 4, page 1027-1038
• ISSN: 2083-5892

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## Abstract

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Let k ≥ 1 be an integer. A signed total Roman k-dominating function on a graph G is a function f : V (G) → {−1, 1, 2} such that Ʃu2N(v) f(u) ≥ k for every v ∈ V (G), where N(v) is the neighborhood of v, and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed total Roman k-dominating functions on G with the property that Ʃdi=1 fi(v) ≤ k for each v ∈ V (G), is called a signed total Roman k-dominating family (of functions) on G. The maximum number of functions in a signed total Roman k-dominating family on G is the signed total Roman k-domatic number of G, denoted by dkstR(G). In this paper we initiate the study of signed total Roman k-domatic numbers in graphs, and we present sharp bounds for dkstR(G). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman k-domatic number of some graphs.

## How to cite

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Lutz Volkmann. "The Signed Total Roman k-Domatic Number Of A Graph." Discussiones Mathematicae Graph Theory 37.4 (2017): 1027-1038. <http://eudml.org/doc/288513>.

@article{LutzVolkmann2017,
abstract = {Let k ≥ 1 be an integer. A signed total Roman k-dominating function on a graph G is a function f : V (G) → \{−1, 1, 2\} such that Ʃu2N(v) f(u) ≥ k for every v ∈ V (G), where N(v) is the neighborhood of v, and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set \{f1, f2, . . . , fd\} of distinct signed total Roman k-dominating functions on G with the property that Ʃdi=1 fi(v) ≤ k for each v ∈ V (G), is called a signed total Roman k-dominating family (of functions) on G. The maximum number of functions in a signed total Roman k-dominating family on G is the signed total Roman k-domatic number of G, denoted by dkstR(G). In this paper we initiate the study of signed total Roman k-domatic numbers in graphs, and we present sharp bounds for dkstR(G). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman k-domatic number of some graphs.},
author = {Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {signed total Roman k-dominating function; signed total Roman k-domination number; signed total Roman k-domatic number.},
language = {eng},
number = {4},
pages = {1027-1038},
title = {The Signed Total Roman k-Domatic Number Of A Graph},
url = {http://eudml.org/doc/288513},
volume = {37},
year = {2017},
}

TY - JOUR
AU - Lutz Volkmann
TI - The Signed Total Roman k-Domatic Number Of A Graph
JO - Discussiones Mathematicae Graph Theory
PY - 2017
VL - 37
IS - 4
SP - 1027
EP - 1038
AB - Let k ≥ 1 be an integer. A signed total Roman k-dominating function on a graph G is a function f : V (G) → {−1, 1, 2} such that Ʃu2N(v) f(u) ≥ k for every v ∈ V (G), where N(v) is the neighborhood of v, and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed total Roman k-dominating functions on G with the property that Ʃdi=1 fi(v) ≤ k for each v ∈ V (G), is called a signed total Roman k-dominating family (of functions) on G. The maximum number of functions in a signed total Roman k-dominating family on G is the signed total Roman k-domatic number of G, denoted by dkstR(G). In this paper we initiate the study of signed total Roman k-domatic numbers in graphs, and we present sharp bounds for dkstR(G). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman k-domatic number of some graphs.
LA - eng
KW - signed total Roman k-dominating function; signed total Roman k-domination number; signed total Roman k-domatic number.
UR - http://eudml.org/doc/288513
ER -

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