Generalizations of Graham's Pebbling Conjecture

Details Generalizations of Graham's Pebbling Conjecture Generalizations of Graham's Pebbling Conjecture

2009-05-19

arxiv.org/abs/0905.3197v1

Mathematics

Combinatorics

18 pages

Scientific paper

We investigate generalizations of pebbling numbers and of Graham's pebbling conjecture that pi(GxH) <= pi(G)pi(H), where pi(G) is the pebbling number of the graph G. 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 show that Sjostrand'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 edge, and we describe an alternate pebbling number for which Graham's conjecture is demonstrably false.

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