Abstract
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f: V (G) → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f 1, f 2, …, f d } of distinct Roman k-dominating functions on G with the property that Σ d i=1 f i (v) ≤ 2 for each v ∈ V (G), is called a Roman k-dominating family (of functions) on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by d kR (G). Note that the Roman 1-domatic number d 1R (G) is the usual Roman domatic number d R (G). In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for d kR (G). In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
Similar content being viewed by others
References
Haynes, T. W., Hedetniemi, S. T., Slater, P. J.: Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998
West, D. B.: Introduction to Graph Theory, Prentice-Hall, Inc, 2000
Zelinka, B.: On k-ply domatic numbers of graphs. Math. Slovaka, 34, 313–318 (1984)
Kämmerling, K., Volkmann, L.: The k-domatic number of a graph. Czech. Math. J., 59(134), 539–550 (2009)
Kämmerling, K., Volkmann, L.: Roman k-domination in graphs. J. Korean Math. Soc., 46, 1309–1318 (2009)
Stewart, I.: Defend the Roman Empire. Sci. Amer., 281(6), 136–139 (1999)
Revelle, C. S., Rosing, K. E.: Defendens imperium romanum: a classical problem in military strategy. Amer. Math. Monthly, 107(7), 585–594 (2000)
Cockayne, E. J., Dreyer, P. A., Jr., Hedetniemi, S. M., et al.: Roman domination in graphs. Discrete Math., 278, 11–22 (2004)
Chambers, E. W., Kinnersley, B., Prince, N., et al.: Extremal problems for Roman domination. SIAM J. Discrete Math., 23, 1575–1586 (2009)
Sheikholeslami, S. M., Volkmann, L.: The Roman domatic number of a graph. Appl. Math. Lett., 23, 1295–1300 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was in part supported by a grant from IPM (Grant No. 89050042)
Rights and permissions
About this article
Cite this article
Sheikholeslami, S.M., Volkmann, L. The Roman k-domatic number of a graph. Acta. Math. Sin.-English Ser. 27, 1899–1906 (2011). https://doi.org/10.1007/s10114-011-0385-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-011-0385-0