On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph G(n, p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u, v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all {u, v} ⊆ V . We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for p not too close to 0 or 1, these bounds are best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if n(1− p) tends to 0, answering a question of O’Rourke.