Generalized Borsuk Graphs
Francisco Martinez-Figueroa

TL;DR
This paper introduces generalized Borsuk graphs based on group actions, linking their chromatic number to topological properties of the space, and provides bounds, conjectures, and probabilistic thresholds for their coloring complexity.
Contribution
It defines $G$-Borsuk graphs, relates their chromatic number to topological invariants, and develops bounds and conjectures, including analysis of random variants.
Findings
Chromatic number determined by $G$-covering number for small $\\epsilon$
Lower bounds derived from $G$-actions on Hom-complexes
Thresholds for random $G$-Borsuk graphs' chromatic number
Abstract
Given a finite group acting freely on a compact metric space , and , we define the -Borsuk graph on by drawing edges whenever there is a non-identity such that . We show that when is small, its chromatic number is determined by the topology of via its -covering number, which is the minimum such that there is a closed cover with for all . We are interested in bounding this number. We give lower bounds using -actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of . We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random -Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology
