Georgia Institute of Technology College of Sciences

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that given any configuration of k pebbles on G and any specified vertex v in V(G), there is a sequence of pebbling moves that sends a pebble to v. We will show that the pebbling number of a graph of diameter four on n vertices is at most 3n/2 + O(1), and this bound is best possible up to an additive constant. This proof, based on a discharging argument and a decomposition of the graph into ''irreducible branches'', generalizes work of Bukh on graphs of diameter three. Further, we prove that the pebbling number of a graph on n vertices with diameter d is at most (2^{d/2} - 1)n + O(1). This also improves a bound of Bukh.