Approximate Solutions of 4-regular Matchstick Graphs with 50-62 Vertices

TL;DR

This paper presents 38 approximate 4-regular matchstick graphs with 50-62 vertices, highlighting the complexity of constructing such graphs with exact unit distances and encouraging further research to find missing cases.

Contribution

The authors provide new approximate solutions for 4-regular matchstick graphs with 50-62 vertices, including previously unresolved cases, illustrating the problem's difficulty.

Findings

38 approximate graphs with 50-62 vertices presented

Graphs contain slight variations from unit length

Highlights the challenge in constructing exact solutions

Abstract

A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. Examples of 4-regular matchstick graphs are currently known for all number of vertices $\geq$ 52 except for 53, 55, 56, 58, 59, 61, and 62. In this article we present 38 different examples with 50 - 62 vertices which contain two, three, or four distances which differ slightly from the unit length. These graphs should show why this subject is so extraordinarily difficult to deal with and should also be an incentive for the interested reader to find solutions for the missing numbers of vertices.