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General penny graphs are at most 43/18-dense

Arsenii Sagdeev

TL;DR

We address the problem of bounding the number of unit-distance edges in penny graphs on $n$ vertices in general position, i.e., obtain $e \le 43n/18$. The authors develop a discharging framework anchored in local structural constraints of the first neighborhood to show $e/n \le 12/5$, then extend the analysis to the second neighborhood via a key structural theorem (Theorem $\text{tn2}$) that every degree-$4$ vertex has an unpopular neighbor outside any kernel, enabling a refined discharging argument with $q=2/9$ to reach $e/n \le 43/18$. Some critical steps rely on computer-assisted verification of complex local configurations. The results close part of the gap between known lower and upper bounds for penny graphs in general position and point toward future work on higher-order neighborhoods and related geometric graphs.

Abstract

We prove that among $n$ points in the plane in general position, the shortest distance occurs at most $43n/18$ times, improving upon the upper bound of $17n/7$ obtained by Tóth in 1997.

General penny graphs are at most 43/18-dense

TL;DR

We address the problem of bounding the number of unit-distance edges in penny graphs on vertices in general position, i.e., obtain . The authors develop a discharging framework anchored in local structural constraints of the first neighborhood to show , then extend the analysis to the second neighborhood via a key structural theorem (Theorem ) that every degree- vertex has an unpopular neighbor outside any kernel, enabling a refined discharging argument with to reach . Some critical steps rely on computer-assisted verification of complex local configurations. The results close part of the gap between known lower and upper bounds for penny graphs in general position and point toward future work on higher-order neighborhoods and related geometric graphs.

Abstract

We prove that among points in the plane in general position, the shortest distance occurs at most times, improving upon the upper bound of obtained by Tóth in 1997.
Paper Structure (5 sections, 21 theorems, 7 equations, 14 figures)

This paper contains 5 sections, 21 theorems, 7 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Structure of the first neighborhood
  3. Structure of the second neighborhood: proof of Theorem \ref{['tmain']}
  4. Proof of Theorem \ref{['tn2']}
  5. Concluding remarks

Key Result

Theorem 1

The maximum number of edges in a matchstick graph on $n$ vertices is $\lfloor 3n-\sqrt{12n-3} \rfloor$.

Figures (14)

  • Figure 1: $|AD'| < |BC|$.
  • Figure 2: $A_1, A$, and $A_4$ are collinear.
  • Figure 3: $A_1, A$, and $A_4$ are collinear.
  • Figure 4: Around a triangle on vertices of degree $5$ without common neighbors, there is a Möbius loop.
  • Figure 5: Around each kernel, there is an apricot.
  • ...and 9 more figures

Theorems & Definitions (37)

  • Theorem 1: LS23
  • Theorem 2: G. Tóth
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 27 more