Diverse beam search to find densest-known planar unit distance graphs

TL;DR

This paper introduces a diverse beam search algorithm to generate dense unit distance graphs, successfully finding all known maximally dense graphs up to 100 vertices and providing new insights into their growth patterns.

Contribution

The paper presents a novel computer algorithm using diverse beam search to find all maximally dense unit distance graphs up to 100 vertices, advancing the understanding of graph density limits.

Findings

Successfully finds all known maximally dense UDGs up to 100 vertices.

Identifies densest UDGs for 15 < n ≤ 30.

Shows similar growth rate of u(n)/n for n > 30.

Abstract

This paper addresses the problem of determining the maximum number of edges in a unit distance graph (UDG) of $n$ vertices using computer search. An unsolved problem of Paul Erd\H{o}s asks the maximum number of edges $u(n)$ a UDG of $n$ vertices can have. Those UDGs that attain $u(n)$ are called "maximally dense." In this paper, we seek to demonstrate a computer algorithm to generate dense UDGs for vertex counts up to at least 100. Via beam search with an added visitation metric, our algorithm finds all known maximally dense UDGs up to isomorphism at the push of a button. In addition, for $15 < n$, where $u(n)$ is unknown, i) the algorithm finds all previously published densest UDGs up to isomorphism for $15 < n \le 30$, and ii) the rate of growth of $u(n)/n$ remains similar for $30 < n$. The code and database of over 60 million UDGs found by our algorithm can be found at https://codeberg.org/zsamboki/dbs-udg.