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Keywords:

  • pebbling;
  • diameter;
  • asymptotic;
  • discharging

Abstract

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 for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We improve on a bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most inline image, and this bound is best possible. Further, we obtain an asymptotic bound of inline image for the pebbling number of graphs of diameter four. Finally, we prove an asymptotic bound for pebbling graphs of arbitrary diameter, namely that the pebbling number for a diameter d graph on n vertices is at most inline image, where inline image is a constant depending upon d. This also improves another bound of Bukh.