Rainbow numbers in planar host graphs

If F is a nonempty set of graphs that contain H as a subgraph, then the rainbow number rb(F,H) is the least t ? N such that every t-coloring of F ? F that uses all t colors contains a rainbow subgraph isomorphic to H. In this paper, we consider rainbow numbers when F = {F} where F is a wheel, a sunflower, or a double-hubbed wheel. Several exact evaluations are determined for various small subgraphs. Implications involving the case where F consists of all plane triangulations of order n are also discussed.