General Bounds on Rainbow Domination Numbers

A k-rainbow dominating function of a graph G is a function f from the vertices V(G) to $${2^{\{1, 2, \dots, k\}}}$$2{1,2,⋯,k} such that, for all $${v \in V(G)}$$v∈V(G), either $${f(v) \neq \emptyset}$$f(v)≠∅ or $${\bigcup_{u \in N[v]} f(u) = \{1, 2, \dots, k\}}$$⋃u∈N[v]f(u)={1,2,⋯,k}. The k-rainbow domination number of a graph G is then defined to be the minimum weight $${w(f) = \sum_{v \in V(G)} |f(v)|}$$w(f)=∑v∈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.