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

For \(n\in\nats\) and \(3\leq k\leq n\) 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 \(\ell\)-gon where \(\ell\geq k\). 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.