Integer-programming bounds on pebbling numbers of cartesian-product graphs

Abstract

Graph pebbling, as introduced by Chung, is a two-player game on a graph G. Player one distributes "pebbles" to vertices and designates a root vertex. Player two attempts to move a pebble to the root vertex via a sequence of pebbling moves, in which two pebbles are removed from one vertex in order to place a single pebble on an adjacent vertex. The pebbling number of a simple graph G is the smallest number pG such that if player one distributes pG pebbles in any configuration, player two can always win. Computing pG is provably difficult, and recent methods for bounding pG have proved computationally intractable, even for moderately sized graphs. Graham conjectured that the pebbling number of the Cartesianproduct of two graphs G and H, denoted G □ H, is no greater than pGpH. Graham's conjecture has been verified for specific families of graphs; however, in general, the problem remains open. This study combines the focus of developing a computationally tractable method for generating good bounds on πG □H, with the goal of providing evidence for (or disproving) Graham's conjecture. In particular, we present a novel integer-programming (IP) approach to bounding πG □ H that results in significantly smaller problem instances compared with existing IP approaches to graph pebbling. Our approach leads to a sizable improvement on the best known bound for πL □L, where L is the Lemke graph. L □ L is among the smallest known potential counterexamples to Graham's conjecture..