Abstract :
Given a distribution of m pebbles on the vertices of a graph G, we allow pebbling moves consisting of taking 2 pebbles off one vertex and placing one of them on an adjacent vertex. We define f(G) to be the least m which guarantees the existence of a sequence of pebbling moves that places a pebble on an arbitrary vertex. It is conjectured that f(G×H)⩽f(G)f(H). In this paper, this is verified in the case where G has Chungʹs 2-pebbling property and H is a complete multi-partite graph, an infinite class of Lemke graphs is found, and new conjectures are raised.
