The intersection of a random geometric graph with an Erdős-Rényi graph

We study the intersection of a random geometric graph with an Erdős-Rényi 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 Erdős-Rényi 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.