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 $\pi(G)$ is the smallest number such that, for any distribution of $\pi(G)$ 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 $\pi(G)$ is provably difficult, and recent methods for bounding $\pi(G)$ have proved computationally intractable, even for moderately sized graphs. Graham conjectured that $\pi(G ~\square~ H) \leq \pi(G) \pi(H)$, where $G ~\square~ H$ 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 $\pi(G ~\square~ H)$, 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 $\pi(L ~\square~ L)$, where $L$ is the Lemke graph, and $L ~\square~ L$ is among the smallest known potential counterexamples to Graham's conjecture.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2019
- DOI:
- 10.48550/arXiv.1905.08683
- arXiv:
- arXiv:1905.08683
- Bibcode:
- 2019arXiv190508683K
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 27 pages, 2 figures, 8 tables