Graham’s pebbling conjecture on product of complete bipartite graphs

Abstract

The pebbling number of a graph G, f(G) , is the least n such that, no matter how n 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. Graham conjectured that for any connected graphs G and H,f(G×H)≤f(G)f(H) . We show that Graham's conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property. As a corollary, Graham's conjecture holds when G and H are complete bipartite graphs.