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.

Abstract. We estimate the likely values of the chromatic and independence numbers of the random [Formula: see text]-uniform [Formula: see text]-regular hypergraph on [Formula: see text] vertices for fixed [Formula: see text], large fixed [Formula: see text], and [Formula: see text].

A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number $\tau(G)$ is at most twice the triangle packing number $\nu(G)$, where the triangle packing number $\nu(G)$ is the maximum size of a set of edge-disjoint triangles in $G$ and the triangle covering number $\tau(G)$ is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erdös--Rényi random graph $G(n,m)$ for all ranges of $m$, closing the “gap” in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and Jinyoung Park.) We employ a random greedy process called the online triangle packing process to produce a triangle packing in $G(n,m)$ and analyze this process by using the differential equations method.