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

Abstract. 

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 off 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 C _m and C _n be cycles. We prove that f(C _m×C _n)≤f(C _m) f(C _n) for all but a finite 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.