Computing Bounds on Product Graph Pebbling Numbers

Abstract Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number π ( G ) is the smallest number such that, for any distribution of π ( G ) pebbles to the vertices of G and choice of root vertex r of G, there exists a sequence of pebbling moves that places a pebble on r. Computing π ( G ) is provably difficult, and recent methods for bounding π ( G ) have proved computationally intractable, even for moderately sized graphs. Graham conjectured that π ( G □ H ) ≤ π ( G ) π ( H ) , where G □ H is the Cartesian product of G and H (1989). While the conjecture has been verified for specific families of graphs, in general it remains open. This study combines the focus of developing a computationally tractable, IP-based method for generating good bounds on π ( G □ H ) , with the goal of shedding light on Graham's conjecture.We provide computational results for a variety of Cartesian-product graphs, including some that are known to satisfy Graham's conjecture and some that are not. Our approach leads to a sizable improvement on the best known bound for π ( L □ L ) , where L is the Lemke graph, and L □ L is among the smallest known potential counterexamples to Graham's conjecture.