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

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.