General Bounds on Rainbow Domination Numbers

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.