Rainbow connectivity of multilayered random geometric graphs

TL;DR

This paper introduces multilayered random geometric graphs and establishes a threshold radius for their rainbow connectivity, extending understanding from Erdős-Rényi models to geometric settings.

Contribution

It defines multilayered random geometric graphs and determines the threshold radius for rainbow connectivity in these models.

Findings

Threshold radius for rainbow connectivity is r(n)=(log n / n)^{(h-1)/(2h)}.

Rainbow connectivity property emerges at this critical radius.

Results extend known thresholds from Erdős-Rényi to geometric graph models.

Abstract

An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from $h\ge 2$ independent random geometric graphs $G(n,r)$ on the unit square. We define an edge-coloring by coloring the edges according to the copy of $G(n,r)$ they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that $r(n)=\left(\frac{\log n}{n}\right)^{\frac{h-1}{2h}}$ is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erd\H{o}s-R\' enyi random model.