Maximum pebbling number of graphs of diameter three

Given a configuration of pebbles on the vertices of a graph G , a pebbling move consists of taking two pebbles off some vertex v and putting one of them back on a vertex adjacent to v . A graph is called pebbleable if for each vertex v there is a sequence of pebbling moves that would place at least one pebble on v . The pebbling number of a graph G is the smallest integer m such that G is pebbleable for every configuration of m pebbles on G . We prove that the pebbling number of a graph of diameter 3 on n vertices is no more than (3/2) n  + O(1), and, by explicit construction, that the bound is sharp. © 2006 Wiley Periodicals, Inc. J Graph Theory