The polychromatic domination number

For a graph \(G\) with a (not necessarily proper) vertex coloring, a set \(D⊆ V(G)\) is a polychromatic dominating set of \(G\) if it is a dominating set and each vertex in \(D\) is a different color. The polychromatic domination number of \(G\), \(ρ(G)\), is the minimum number of colors such that, for any \(ρ(G)\)-coloring (with exactly \(ρ(G)\) colors) of the vertices of \(G\), there exists a polychromatic dominating set of \(G\). This paper begins the exploration of the polychromatic domination number. In particular we give tight upper and lower bounds for \(ρ(G)\) both of which are functions of the minimum degree of \(G\).