The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles o one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H , f(G H) f(G)f(H). Let Cm and Cn be cycles. We prove that f(Cm Cn) f(Cm)f(Cn) for all but a nite number of possible cases. We also prove that f(G T ) f(G)f(T ) when G has the 2-pebbling property and T is any tree.