Constructions for the Optimal Pebbling of Grids

Abstract

In [6] 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.