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. A graph is called pebbleable if for each vertex v there is a sequence of pebbling moves so that at least one pebble can be placed on vertex v. The pebbling number of a graph G is the smallest integer k such that G is pebbleable given any configuration of k pebbles on G. We improve on the bound of Bukh by showing that the pebbling number of a graph of diameter 3 on n vertices is at most the floor of 3n/2 + 2, and this bound is best possible. We give an alternative proof that the pebbling number of a graph of diameter 2 on n vertices is at most n + 1. This is joint work with Noah Streib and Carl Yerger.