Constructions for the optimal pebbling of grids

TL;DR

This paper disproves a conjecture about the maximum covering ratio in pebbling on infinite grids by providing a counterexample with ratio 3.5, and discusses challenges in establishing upper bounds.

Contribution

It presents a counterexample to a conjectured maximum covering ratio and analyzes difficulties in proving upper bounds for pebbling on grids.

Findings

Counterexample with covering ratio 3.5 disproves the conjecture.

Highlights issues in existing proof claiming a maximum ratio of 6.75.

Discusses new pebble distributions and challenges in the problem.

Abstract

In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics] the authors conjecture that if every vertex of an infinite square grid is reachable from a pebble distribution, then the covering ratio of this distribution is at most $3.25$. First we present such a distribution with covering ratio $3.5$, disproving the conjecture. The authors in the above paper also claim to prove that the covering ratio of any pebble distribution is at most $6.75$. The proof contains some errors. We present a few interesting pebble distributions that this proof does not seem to cover and highlight some other difficulties of this topic.