Pebbling Graphs of Fixed Diameter

Given a configuration of indistinguishable pebbles on the vertices of a connected graph G on n vertices, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The m-pebbling number of a graph G, πm(G), is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least m pebbles on v. When m = 1, it is simply called the pebbling number of a graph. We prove that if G is a graph of diameter d and k,m ≥ 1 are integers, then πm(G) ≤ f(k)n + 2m + (2(2 − 1) − f(k))domk(G), where domk(G) denotes the size of the smallest distance k dominating set, that is the smallest subset of vertices such that every vertex is at most distance k from the set, and, f(k) = (2−1)/k. This generalizes the work of Chan and Godbole [4] who proved this formula for k = m = 1. As a corollary, we prove that πm(G) ≤ f(dd/2e)n+O(m+ √ n lnn). Furthermore, we prove that if d is odd, then πm(G) ≤ f(dd/2e)n + O(m), which in the case of m = 1 answers for odd d, up to a constant additive factor, a question of Bukh [3] about the best possible bound on the pebbling number of a graph with respect to its diameter.