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.
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Notes
- 1.
Our bound carries the caveat that it was obtained by a solver that employs floating point arithmetic.
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Kenter, F., Skipper, D. (2018). Integer-Programming Bounds on Pebbling Numbers of Cartesian-Product Graphs. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_46
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