On some non-rigid unit distance patterns

TL;DR

This paper investigates bounds on non-rigid unit distance patterns, including paths and cycles on a sphere and 3-regular graphs in 3D, extending the Erdős unit distance problem to new geometric configurations.

Contribution

It provides sharp bounds for unit distance paths and cycles on a specific sphere and explores properties of 3-regular unit distance graphs in three-dimensional space.

Findings

Sharp bounds on unit distance paths and cycles on the sphere of radius 1/√2.

Results on 3-regular unit distance graphs in 3D space.

Extension of Erdős unit distance problem to non-rigid patterns.

Abstract

A recent generalization of the Erd\H{o}s Unit Distance Problem, proposed by Palsson, Senger and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in $3$-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius $1/\sqrt{2}$. We also consider a similar problem about $3$-regular unit distance graphs in $\mathbb{R}^3$.