The Borsuk number of a graph

TL;DR

This paper explores the Borsuk problem within graph theory, analyzing its complexity, providing exact solutions, and establishing bounds for different graph partitioning scenarios.

Contribution

It introduces a graph-based formulation of the Borsuk problem, examines two settings for graph diameters, and offers complexity results and bounds.

Findings

Presented complexity results for the graph Borsuk problem.

Computed exact solutions for specific cases.

Established upper bounds for the problem parameters.

Abstract

The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject of much attention over the years, and research on variants of the problem continues nowadays in a plethora of directions. In this work, we propose a formulation of the problem in the context of graphs. Depending on how the graph is partitioned, we consider two different settings dealing either with the usual notion of diameter in abstract graphs, or with the diameter in the context of continuous graphs, where all points along the edges, instead of only the vertices, must be taken into account when computing distances. We present complexity results, exact computations and upper bounds on the parameters associated to the problem.