Distributed Colouring with 4/3 chi Colours for Hyperbolic Random Graphs

2026-07-22

AI Summary

This work addresses the problem of efficient distributed vertex coloring in hyperbolic random graphs. The authors propose the Sequential Radial Colouring algorithm, which leverages radial layering and local computation within the CONGEST model to significantly reduce both the number of colors used and the number of communication rounds. Their main contributions include the first high-probability near-optimal coloring for hyperbolic random graphs using only \(4/3\,\chi\) colors, where \(\chi\) denotes the chromatic number. They further introduce algorithmic variants achieving \(4/3\,\chi\) colors in \(O((\log\log n)^2)\) rounds, \(O(\chi\log\log n)\) colors in \(O(\log\log n)\) rounds, or \(\chi^{1+\varepsilon}\) colors in a constant number of rounds. Additionally, they refine the theoretical trade-off between slack and round complexity in clique-based coloring.

Abstract

We study distributed vertex colouring on Hyperbolic Random Graphs (HRGs), a geometric random graph model capturing key structural features of real-world networks. This provides a natural setting for analysing distributed algorithms beyond worst-case general graphs. We introduce Sequential Radial Colouring, a CONGEST algorithm using only efficient local computation. The algorithm achieves a near-optimal palette, colouring HRGs with $\frac{4}{3}χ$ colours and running in $O((\log\log n)^2)$ rounds a.a.s. We also give a variant that speeds this up to $O(\log\log n)$ rounds a.a.s., at the price of using $O(χ\log\log n)$ colours. Finally, for every constant $\varepsilon>0$, it runs in $O(1)$ rounds a.a.s. when $χ^{1+\varepsilon}$ colours are used. This greatly reduces the number of colours over the previous constant-round algorithm of Maus and Ruff (SODA 2026) by a factor of at least $n^{1/6}$. Our analysis contains a phase in which we consider a classical randomised colouring protocol on a (large) clique of the graph. We also delve deeper into this part of the analysis and improve upon previous results for colouring a clique $C$, bounding the number of rounds required as a function of the additive slack $s = |Ψ| - χ$, where $Ψ$ is the set of colours used. In particular, constant-round colouring is possible if and only if $s=|C|^{1+Ω(1)}$, while $s=|C|/\log |C|$ already gives the optimal $Θ(\log\log |C|)$ round complexity.

Problem

• distributed colouring
• hyperbolic random graphs
• vertex colouring
• CONGEST model
• round complexity

Innovation

• Hyperbolic Random Graphs
• Distributed Vertex Colouring
• Sequential Radial Colouring
• CONGEST Model
• Clique Colouring