Generalizations of Graham’s pebbling conjecture

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that π(G□H)≤π(G)π(H)π(G□H)≤π(G)π(H), where π(G)π(G) is the pebbling number of the graph GG. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that shows that Sjöstrand’s theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the weight of the edge and we describe an alternate pebbling number for which Graham’s conjecture is demonstrably false.