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Keywords:

  • graph pebbling;
  • dominating sets

Abstract

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, inline image, 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 inline image, it is simply called the pebbling number of a graph. We prove that if G is a graph of diameter d and inline image are integers, then inline image, where inline image 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, inline image. This generalizes the work of Chan and Godbole (Discrete Math 208 (2008), 15–23) who proved this formula for inline image. As a corollary, we prove that inline image. Furthermore, we prove that if d is odd, then inline image, which in the case of inline image answers for odd d, up to a constant additive factor, a question of Bukh (J Graph Theory 52 (2006), 353–357) about the best possible bound on the pebbling number of a graph with respect to its diameter.