Regular matchstick graphs on the sphere

TL;DR

Problem addressed: classify all $5$-regular matchstick graphs on the sphere. Approach: reinterpret as contact graphs of equal spherical caps and apply a spherical discharging method to enforce $5$-regularity and restrict face types; show non-edge distances exceed the edge length. Result: complete classification as Robinson's five cap-contact graphs with $|V|\in\{12,24,60,48,120\}$, with a stronger form under a minimum-degree assumption. Significance: connects discrete graph theory on the sphere to classical cap-packings, yielding concrete geometric realizations on the sphere and clarifying analogous results in elliptic, hyperbolic, and Euclidean geometries and informing related classifications.

Abstract

We show that the $5$-regular matchstick graphs on the sphere are exactly the five $5$-regular contact graphs of congruent caps on the sphere found by R. M. Robinson (1969).

Figures (5)

• Figure 1: Icosahedron gives a $5$-regular matchstick graph on $12$ vertices on the sphere
• Figure 2: Snub cube gives a $5$-regular matchstick graph on $24$ vertices on the sphere
• Figure 3: Robinson's $5$-regular matchstick graph on $48$ vertices on the sphere
• Figure 4: Snub icosahedron gives a $5$-regular matchstick graph on $60$ vertices on the sphere
• Figure 5: Robinson's $5$-regular matchstick graph on $120$ vertices on the sphere

Theorems & Definitions (2)

• Theorem
• proof