For $n\in\mathbb{N}$ and $4\leq k\leq n$ we compute the exact value of $E_k(n)$, the maximum number of edges of a simple plane graph on n vertices, where each vertex bounds an $\ell$-gon where $\ell\geq k$. The lower bound of $E_k(n)$ is obtained by explicit construction, while the matching upper bound is obtained by solving an integer program by inspection/picture. We then use this result to conjecture the maximum number of edges of a nonflowerable coin graph on n vertices. A flower is a coin graph representation of the wheel graph. A collection of coins or discs in the Euclidean plane is nonflowerable if no flower can be formed by coins from the collection.

We give nontrivial boundsfor the inductiveness or degeneracy of power graphs G k of a planar graph G. This implies bounds for the chromatic number as well, since the inductiveness naturally relates to a greedy algorithm for vertex-coloring the given graph. The inductiveness moreover yields bounds for the choosability of the graph. We show that the inductiveness of a square of a planar graph G is at most $\lceil 9\Delta /5 \rceil$, for the maximum degree $\Delta$ sufficiently large, and that it is sharp. In general, we show for a fixed integer $k\geq1$ the inductiveness, the chromatic number, and the choosability of G k to be $O(\Delta^{\lfloor k/2 \rfloor})$, which is tight.