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 $\pi _G$ such that if player one distributes $\pi _G$ pebbles in any configuration, player two can always win. Computing $\pi _G$ is provably difficult, and recent methods for bounding $\pi _G$ have proved computationally intractable, even for moderately sized graphs.

Graham conjectured that the pebbling number of the Cartesian-product of two graphs G and H, denoted $G\,\square \,H$, is no greater than $\pi _G \pi _H$. 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 $\pi _{G \,\square \, H}$, with the goal of providing evidence for (or disproving) Graham’s conjecture. In particular, we present a novel integer-programming (IP) approach to bounding $\pi _{G \,\square \, H}$ 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 $\pi _{L \,\square \, L}$, where L is the Lemke graph. $L\,\square \, L$ is among the smallest known potential counterexamples to Graham’s conjecture.

F. Kenter—Partially funded by the Naval Academy Research Council and by NSF Grant DMS-1719894.

D. Skipper—Partially funded by the Naval Academy Research Council.