On the maximum number of edges of non-flowerable coin graphs

For $n\in \text{nats}$ and $3\le k\le n$ we compute the exact value of $E_k(n)$, the maximum number of edges of a simple planar graph on $n$ vertices where each vertex bounds an $\mathrm{\ell }$-gon where $\mathrm{\ell }\ge k$. The lower bound of $E_k(n)$ is obtained by explicit construction, and the matching upper bound is obtained by using Integer Programming (IP.) We then use this result to conjecture the maximum number of edges of a non-flowerable coin graph on $n$ vertices. A {\em flower} is a coin graph representation of the wheel graph. A collection of coins or discs in the Euclidean plane is {\em non-flowerable} if no flower can be formed by coins from the collection.