Colouring random Hasse diagrams and box-Delaunay graphs

TL;DR

This paper studies the chromatic and independence numbers of random Hasse diagrams and box-Delaunay graphs in high dimensions, providing asymptotic bounds that extend previous results and resolve a conjecture.

Contribution

It establishes precise asymptotic bounds for the chromatic and independence numbers of these graphs in random settings, extending and sharpening prior work.

Findings

Chromatic number typically grows as (log n)^{d-1+o(1)}

Independence number typically decreases as n/( ext{log n})^{d-1+o(1)}

Results resolve a conjecture of Tomon about box-Delaunay graphs

Abstract

Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to axis-parallel boxes (where we put an edge between $u,v\in P$ whenever there is an axis-parallel box containing $u,v$ and no other points of $P$). In each of these two closely related settings, we show that the chromatic number of $G$ is typically $(\log n)^{d-1+o(1)}$ and the independence number of $G$ is typically $n/(\log n)^{d-1+o(1)}$. When $d=2$, we obtain bounds that are sharp up to constant factors: the chromatic number is typically of order $\log n/\log\log n$ and the independence number is typically of order $n\log\log n/\log n$. These results extend and sharpen previous bounds by Chen, Pach, Szegedy and Tardos. In addition, they provide new bounds on the largest possible chromatic number (and lowest possible independence number) of a $d$-dimensional box-Delaunay graph or Hasse diagram, in particular resolving a conjecture of Tomon.