The Connectivity Threshold for Dense Graphs

Consider a random graph model where there is an underlying simple graph G = (V, E), and each edge is sampled independently with probability p ∊ [0, 1]. What is the smallest value of p such that the resulting graph Gp is connected with constant probability? This is a well-studied question for special classes of graphs, such as complete graphs and hypercubes. For instance, when G is the complete graph, we want the connectivity threshold for the Erdős-Rényi G(n, p) model: here the answer is known to be . However, the problem is not well-understood for more general graph classes. We first investigate this connectivity threshold problem for “somewhat dense” graphs. We show that for any and any δ-regular, δ-edge-connected graph G, the random graph Gp for is connected with probability , generalizing upon the case when G is the complete graph. Our proof also bounds the number of approximate mincuts in such a dense graph, which may be of independent interest. Next, for a general graph G with edge connectivity λ, we define an explicit parameter βG ∊ (0, 2 ln n], based on the number of approximate mincuts, and show that there is a sharp transition in the connectivity of G at p = 1 – exp(βG/λ). Moreover, we show that the width of this transition is an additive O(ln λ/λ) term; this improves upon Margulis' classical result bounding the width of the threshold by .