The intersection of a random geometric graph with an Erd\H{o}s-R\'enyi graph

TL;DR

This paper investigates the properties of the intersection of a random geometric graph and an Erdős-Rényi graph, analyzing how key graph parameters behave when both connection radius and edge probability tend to zero.

Contribution

It extends previous work by studying the intersection graph with both radius and edge probability diminishing, providing new insights into its structural properties.

Findings

Analyzed clique and independence numbers in the intersection graph.

Studied connectivity and Hamiltonicity under shrinking parameters.

Provided asymptotic behavior of chromatic number and diameter.

Abstract

We study the intersection of a random geometric graph with an Erd\H{o}s-R\'enyi graph. Specifically, we generate the random geometric graph $G(n, r)$ by choosing $n$ points uniformly at random from $D=[0, 1]^2$ and joining any two points whose Euclidean distance is at most $r$. We let $G(n, p)$ be the classical Erd\H{o}s-R\'enyi graph, i.e. it has $n$ vertices and every pair of vertices is adjacent with probability $p$ independently. In this note we study $G(n, r, p):=G(n, r) \cap G(n, p)$. One way to think of this graph is that we take $G(n, r)$ and then randomly delete edges with probability $1-p$ independently. We consider the clique number, independence number, connectivity, Hamiltonicity, chromatic number, and diameter of this graph where both $p(n)\to 0$ and $r(n)\to 0$; the same model was studied by Kahle, Tian and Wang (2023) for $r(n)\to 0$ but $p$ fixed.