Thresholds for random distributions on graph sequences with applications to pebbling

Let G = {G1, G2,..., Gn .... } be a sequence of graphs with Gn having n vertices and having a random distribution of t(n) pebbles to its vertices. If s ≥ 2 is an integer, the event that Gn has s or more vertices with two or more pebbles has threshold t(n) = Θ(√n). If t(n) = c√n, then the limiting distribution for the number of vertices with multiple pebbles is Poisson(c2). The threshold for the event that Gn has at least one vertex with s or more pebbles is t(n) = Θ(n(s-1)/s). These results are used to establish new bounds for thresholds for pebbling on sequences of graphs with bounded diameters. If for some d, diameter (Gn) ≤ d for all n, and if for some p ∈ (0, 1], maximum degree (Gn) ⊆ Ω(np), then the threshold th(G) for the solvability of G is in O(n1-0.5p).