Year: 2013
Type: preprint
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 Vázsonyi'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.
Source: arXiv (Cornell University)
Author: Kupavskii, Andrey
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2951898184