Abstract
A k-rainbow dominating function of a graph G is a function f from the vertices V(G) to \({2^{\{1, 2, \dots, k\}}}\) such that, for all \({v \in V(G)}\), either \({f(v) \neq \emptyset}\) or \({\bigcup_{u \in N[v]} f(u) = \{1, 2, \dots, k\}}\). The k-rainbow domination number of a graph G is then defined to be the minimum weight \({w(f) = \sum_{v \in V(G)} |f(v)|}\) of a k-rainbow dominating function. In this work, we prove sharp upper bounds on the k-rainbow domination number for all values of k. Furthermore, we also consider the problem with minimum degree restrictions on the graph.
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S. Fujita is supported by the Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists (B) (20740095).
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Fujita, S., Furuya, M. & Magnant, C. General Bounds on Rainbow Domination Numbers. Graphs and Combinatorics 31, 601–613 (2015). https://doi.org/10.1007/s00373-013-1394-9
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DOI: https://doi.org/10.1007/s00373-013-1394-9