# Upper Bounds on the Signed Total (K, K)-Domatic Number of Graphs

• Volume: 35, Issue: 4, page 641-650
• ISSN: 2083-5892

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## Abstract

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Let G be a graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and Σx∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that Σdi=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed total (k, k)- domatic number, in particular for regular graphs.

## How to cite

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Lutz Volkmann. "Upper Bounds on the Signed Total (K, K)-Domatic Number of Graphs." Discussiones Mathematicae Graph Theory 35.4 (2015): 641-650. <http://eudml.org/doc/276003>.

@article{LutzVolkmann2015,
abstract = {Let G be a graph with vertex set V (G), and let f : V (G) → \{−1, 1\} be a two-valued function. If k ≥ 1 is an integer and Σx∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set \{f1, f2, . . . , fd\} of distinct signed total k-dominating functions on G with the property that Σdi=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed total (k, k)- domatic number, in particular for regular graphs.},
author = {Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {signed total (k; k)-domatic number; signed total k-dominating function; signed total k-domination number; regular graphs; signed total -domatic number; signed total -dominating function; signed total -domination number},
language = {eng},
number = {4},
pages = {641-650},
title = {Upper Bounds on the Signed Total (K, K)-Domatic Number of Graphs},
url = {http://eudml.org/doc/276003},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Lutz Volkmann
TI - Upper Bounds on the Signed Total (K, K)-Domatic Number of Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 4
SP - 641
EP - 650
AB - Let G be a graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and Σx∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that Σdi=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed total (k, k)- domatic number, in particular for regular graphs.
LA - eng
KW - signed total (k; k)-domatic number; signed total k-dominating function; signed total k-domination number; regular graphs; signed total -domatic number; signed total -dominating function; signed total -domination number
UR - http://eudml.org/doc/276003
ER -

## References

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1. [1] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
2. [2] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Ed(s), Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998). Zbl0883.00011
3. [3] M.A. Henning, Signed total domination in graphs, Discrete Math. 278 (2004) 109-125. doi:10.1016/j.disc.2003.06.002[Crossref]
4. [4] M.A. Henning, On the signed total domatic number of a graph, Ars Combin. 79 (2006) 277-288. Zbl1164.05421
5. [5] S.M. Sheikholeslami and L. Volkmann, Signed total (k, k)-domatic number of a graph, AKCE Int. H. J. Graphs Comb. 7 (2010) 189-199. Zbl1248.05143
6. [6] C. Wang, The signed k-domination numbers in graphs, Ars Combin. 106 (2012) 205-211. Zbl1289.05363
7. [7] B. Zelinka, Signed total domination number of a graph, Czechoslovak. Math. J. 51 (2001) 225-229. doi:10.1023/A:1013782511179 [Crossref] Zbl0977.05096

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