2013/06/17 by Kupavskii, Andrey
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1306.3910
A diameter graph in \mathbb Rd is a graph, whose set of vertices is a finite subset of \mathbb Rd 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 R4 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/√ 2. We prove Schur's conjecture for diameter graphs in \mathbb R4. We also establish the maximum number of triangles a diameter graph in \mathbb R4 can have, showing that the extremum is attained only on specific Lenz configurations.