Penny graphs in the hyperbolic plane

Abstract

We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a spiral construction of the tiling of the plane with equilateral triangles (Harborth 1974), with a similar result in the hyperbolic plane for the values of $d$ corresponding to the order-$k$ triangular tilings (Bowen 2000). We present various upper and lower bounds for $e_d(n)$ for all values of $d>0$. In particular, we prove that if $d>0.66114\dots$ except for $d=0.76217\dots$, then the number of touching pairs is less than the one coming from a spiral construction in the order-$7$ triangular tiling, which we conjecture to be extremal. We also give a lower bound $e_d(n)>(2+{\epsilon }_d)n$ where ${\epsilon }_d>1$ for all $d>0$.

16 pages, 6 figures