# Integer-Programming Bounds on Pebbling Numbers of Cartesian-Product Graphs

• Franklin Kenter
• Daphne Skipper
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

## 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.

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