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Minimal 3-regular Penny Graph

Alexander Karabegov, Tanya Khovanova

TL;DR

The paper addresses the minimal size of a $3$-regular penny graph, showing that such graphs must have at least $16$ vertices and constructing a valid $16$-vertex example. It employs building-block constructions from unit-distance components (triangles and rhombuses) and a boundary-angle analysis to bound the possible structures, culminating in a rigorous lower bound and a concrete minimal instance. The work clarifies the structure of penny graphs in the $3$-regular case and confirms Karabegov's conjecture on minimal vertex count, with implications for unit-distance and matchstick graph classifications.

Abstract

We prove that a 3-regular penny graph has at least 16 vertices and show that such a graph with 16 vertices exists.

Minimal 3-regular Penny Graph

TL;DR

The paper addresses the minimal size of a -regular penny graph, showing that such graphs must have at least vertices and constructing a valid -vertex example. It employs building-block constructions from unit-distance components (triangles and rhombuses) and a boundary-angle analysis to bound the possible structures, culminating in a rigorous lower bound and a concrete minimal instance. The work clarifies the structure of penny graphs in the -regular case and confirms Karabegov's conjecture on minimal vertex count, with implications for unit-distance and matchstick graph classifications.

Abstract

We prove that a 3-regular penny graph has at least 16 vertices and show that such a graph with 16 vertices exists.
Paper Structure (5 sections, 4 theorems, 2 equations, 8 figures)

This paper contains 5 sections, 4 theorems, 2 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Building Blocks
  4. Boundary
  5. Acknowledgments

Key Result

Proposition 2

The closest distance is unique within the minimal set.

Figures (8)

  • Figure 1: A solution with 24 points
  • Figure 2: A solution with 16 points
  • Figure 3: Two unit distance graphs with a matchstick graph on the right
  • Figure 4: Triangle and rhombus
  • Figure 5: A construction with $3$ rhombuses
  • ...and 3 more figures

Theorems & Definitions (8)

  • Conjecture 1: Karabegov’s Conjecture
  • Proposition 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof