Generalizations of Graham's Pebbling Conjecture

TL;DR

This paper explores generalizations of Graham's pebbling conjecture, introduces new methods to approach it, and presents counterexamples that challenge existing assumptions about pebbling numbers and related theorems.

Contribution

It develops new techniques for attacking Graham's pebbling conjecture and introduces an alternative pebbling number where the conjecture fails.

Findings

Certain conjectures imply others, indicating a hierarchy of pebbling conjectures.

Counterexamples show that Sjostrand's theorem does not hold under edge-dependent transfer costs.

An alternative pebbling number demonstrates Graham's conjecture can be false.

Abstract

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.