Thresholds for colouring the random Borsuk graph

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 $>π-α$ where the parameter $α=α(n)$ 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 $α(n) = c \cdot n^{-1/d}$, 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 $n$".

Figures (3)

• Figure 1: Visualization of the random Borsuk graph with $d=2, n=32, \alpha=1$ and a proper vertex colouring with three colours.
• Figure 2: Schematic of the event $E_{\ell,k,R}$.
• Figure 3: Relation between $H$ and $R$ for $r>1$ (left) and $r\leq 1$ (right).

Theorems & Definitions (59)

• Theorem 1
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• Lemma 1
• Lemma 2
• Proposition 1
• Lemma 3
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• Lemma 5