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 ) ∩ 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 ) → 0 and r ( n ) → 0 ; the same model was studied by Kahle et al. (2023) for r ( n ) → 0 but p fixed.