Rainbow Numbers in Planar Host Graphs

Abstract

If $\mathcal{F}$ is a nonempty set of graphs that contain $H$ as a subgraph, then the rainbow number ${\rm rb}(\mathcal{F}, H)$ is the least $t\in \mathbb{N}$ such that every $t$-coloring of $F\in\mathcal{F}$ that uses all $t$ colors contains a rainbow subgraph isomorphic to $H$.  In this paper, we consider rainbow numbers when $\mathcal{F}=\{F\}$ 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 $\mathcal{F}$ consists of all plane triangulations of order $n$ are also discussed.