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• # Artykuł - szczegóły

## Discussiones Mathematicae Graph Theory

2015 | 35 | 4 | 641-650

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

EN

### Abstrakty

EN
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.

EN

641-650

wydano
2015-11-01
otrzymano
2013-08-20
poprawiono
2014-04-05
zaakceptowano
2015-01-19
online
2015-11-10

### Twórcy

autor
• Lehrstuhl II für Mathematik RWTH Aachen University 52056 Aachen, Germany

### Bibliografia

• [1] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).
• [2] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Ed(s), Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998).
• [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] M.A. Henning, On the signed total domatic number of a graph, Ars Combin. 79 (2006) 277-288.
• [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.
• [6] C. Wang, The signed k-domination numbers in graphs, Ars Combin. 106 (2012) 205-211.
• [7] B. Zelinka, Signed total domination number of a graph, Czechoslovak. Math. J. 51 (2001) 225-229. doi:10.1023/A:1013782511179 [Crossref]