Diameter graphs in $\mathbb R^4$

TL;DR

This paper investigates extremal properties of diameter graphs in four-dimensional space and on a sphere, proving conjectures and establishing maximum triangle counts with specific configurations.

Contribution

It proves analogues of Vázsonyi's and Borsuk's conjectures for diameter graphs on spheres and confirms Schur's conjecture in -dimensional space, identifying extremal configurations.

Findings

Proved an analogue of Ve1zsonyi's and Borsuk's conjecture for diameter graphs on a sphere.

Confirmed Schur's conjecture for diameter graphs in -dimensional space.

Determined the maximum number of triangles in diameter graphs in -space, attained only on Lenz configurations.

Abstract

A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $\mathbb R^4$ and on a three-dimensional sphere. We prove an analogue of V\'azsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $1/\sqrt 2$. We prove Schur's conjecture for diameter graphs in $\mathbb R^4.$ We also establish the maximum number of triangles a diameter graph in $\mathbb R^4$ can have, showing that the extremum is attained only on specific Lenz configurations.