Thresholds for colouring the random Borsuk graph

TL;DR

This paper investigates the chromatic number thresholds of the random Borsuk graph on a sphere, revealing constant average degree regimes for colorability transitions and establishing sharp thresholds related to percolation theory.

Contribution

It establishes that colorability transitions occur at constant average degree levels for each k, and identifies explicit sharp thresholds for these transitions, linking them to continuum percolation.

Findings

Colorability transition for each k occurs at constant average degree.

Sharp threshold for k=2 at  \, n^{-1/d}, linked to continuum AB percolation.

Almost all n exhibit sharp thresholds for k=3,+1.

Abstract

We consider the chromatic number of the random Borsuk graph. The random Borsuk graph is obtained by sampling $n$ points i.i.d. uniformly at random on the $d$-dimensional sphere $S^d$, and joining a pair of points by an edge whenever their geodesic distance is $>\pi-\alpha$ where the parameter $\alpha=\alpha(n)$ may depend on $n$. Kahle and Martinez-Figueroa have shown that the switch from being $(d+1)$-colourable to needing $\geq d+2$ colours occurs in the regime where the average degree is of logarithmic order. We show that for each $2\leq k\leq d$, the switch from being $k$-colourable to needing $> k$ colours occurs in the regime when the average degree is constant. What is more, we show that for $k=2$ there is a sharp threshold of the form $\alpha(n) = c \cdot n^{-1/d}$, where the constant $c$ can be expressed in terms of the critical intensity for continuum AB percolation on $\mathbb{R}^d$. For $k=3,\dots,d+1$ we show that there is a sharp threshold for "almost all $n$".