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Generalized Borsuk Graphs

Francisco Martinez-Figueroa

Abstract

Given a finite group $G$ acting freely on a compact metric space $M$, and $ε>0$, 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 $d(x,gy)\leqε$. We show that when $ε$ 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 $ε$ 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.

Generalized Borsuk Graphs

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 that guarantee that the chromatic number is still that of the whole -Borsuk graph. Our results are tight (up to a constant) when the -index and dimension of coincide.

Paper Structure

This paper contains 18 sections, 27 theorems, 74 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Definitions and Results
  4. Notation:
  5. Properties of $\bm{\text{cov}_G}$
  6. Lower Bound
  7. $\bm{\text{Hom}(K_m,\cdot)}$ as $\bm{G}$-space.
  8. Proof of Theorem \ref{['thm:lower_bound_cov']}
  9. Upper Bounds
  10. Random $\bm{G}$-Borsuk Graphs
  11. Some Geometric Lemmas
  12. Other Preliminary Results
  13. Proof of Theorem \ref{['thm:chromatic_random']}
  14. Dense Case
  15. Sparse Case
  16. ...and 3 more sections

Key Result

Theorem 2.7

Figures (1)

  • Figure 1: Optimal $G$-coverings for 2-dimensional classifying spaces of $G=\mathbb{Z}_3,\mathbb{Z}_4, \mathbb{Z}_5$ and $\mathbb{Z}_6$

Theorems & Definitions (62)

  • Definition 2.1: $G$-spaces
  • Definition 2.2: Simplicial $G$-spaces
  • Definition 2.3: $G$-maps
  • Definition 2.4: Finite Classifying Spaces
  • Definition 2.6: $G$-index
  • Theorem 2.7: Matousek2008
  • Definition 2.8
  • Definition 2.9
  • Definition 3.1: $G$-Borsuk Graphs
  • Definition 3.2: $G$-covering number
  • ...and 52 more