The Roman <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph |
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Authors: | Seyed Mahmoud Sheikholeslami Lutz Volkmann |
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Institution: | 1.Department of Mathematics,Azarbaijan University of Tarbiat Moallem,Tabriz,I. R. Iran;2.School of Mathematics, Institute for Research in Fundamental Sciences (IPM),Tehran,I. R. Iran;3.Lehrstuhl II für Mathematik,RWTH Aachen University,Aachen,Germany |
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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
i=1
d
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. |
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Keywords: | Roman domination number Roman domatic number Roman k-domination number Ro- man k-domatic number |
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