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

TL;DR

This paper determines the maximum edges in certain planar graphs with polygonal bounds, uses integer programming for bounds, and conjectures limits for non-flowerable coin graphs, contributing to geometric graph theory.

Contribution

It provides exact maximum edge counts for specific planar graphs and introduces a conjecture on non-flowerable coin graphs based on these results.

Findings

Exact values of E_k(n) for various n and k

Explicit constructions achieving the lower bounds

Integer programming used to establish upper bounds

Abstract

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.