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.