Extreme problems of circle packings on a sphere and irreducible contact graphs

TL;DR

This paper advances the understanding of optimal circle packings on spheres by enumerating irreducible contact graphs, enabling solutions to classical extremal problems like Tammes and maximal contacts.

Contribution

It provides a comprehensive enumeration of irreducible contact graphs on the sphere, facilitating solutions to various extremal packing problems.

Findings

Enumeration of all irreducible contact graphs for N<12

Application to Tammes problem and maximal contacts

Resolution of several classical packing problems

Abstract

Recently, we enumerate up to isometry, all locally rigid circle packings on the unit sphere with number of circles N<12. This problem is equivalent to the enumeration of irreducible contact graphs. In this paper we show that by using the list of irreducible graphs can solve various problems of extreme packings such as the Tammes problem for the sphere and the projective plane, the maximal contacts problem, Danzer's and other problems on irreducible contact graphs.