On pebbling threshold functions for graph sequences Original Research Article

Given a connected graph G, and a distribution of t pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number t, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs G=(G1,G2,…,Gn,…), where Gn has n vertices, is any function t0(n) such that almost all distributions of t pebbles are solvable when t⪢t0, and such that almost none are solvable when t⪡t0. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.