Generalized Borsuk Graphs

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 $G$ acting freely on a compact metric space $M$, and $\epsilon>0$, we define the $G$-Borsuk graph on $M$ by drawing edges $x\sim y$ whenever there is a non-identity $g\in G$ such that $d(x,gy)\leq\epsilon$. We show that when $\epsilon$ is small, its chromatic number is determined by the topology of $M$ via its $G$-covering number, which is the minimum $k$ such that there is a closed cover $M=F_1\cup\dots\cup F_k$ with $F_i\cap g(F_i)=\emptyset$ for all $g\in G\setminus\{1\}$. We are interested in bounding this number. We give lower bounds using $G$-actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of $M$. We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random $G$-Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for $\epsilon$ that guarantee that the chromatic number is still that of the whole $G$-Borsuk graph. Our results are tight (up to a constant) when the $G$-index and dimension of $M$ coincide.