The Borsuk number of a graph

2026-04-13

AI Summary

This work introduces the classical Borsuk partition problem into a graph-theoretic framework, investigating how to partition a graph into subsets such that each subset has diameter strictly smaller than that of the original graph. It formally defines the Borsuk number of a graph under both discrete (considering only vertex distances) and continuous (including all points along edges) models, and conducts a systematic analysis by integrating tools from graph theory, combinatorial geometry, and computational complexity. The main contributions include the first formal definition of the Borsuk number for graphs, derivation of exact values and tight upper bounds, and proofs establishing that determining or constructing an optimal partition is NP-hard in several settings, thereby extending the frontier of interdisciplinary research between combinatorial geometry and graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Borsuk number
• graph diameter
• combinatorial geometry
• graph partitioning
• continuous graphs

Innovation

• Borsuk number
• graph diameter
• combinatorial geometry
• continuous graphs
• graph partitioning