The optimal pebbling of spindle graphs

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that for some distribution of m pebbles on G, one pebble can be moved to any vertex of G by a sequence of pebbling moves. Let Pk be the path on k vertices. Snevily defined the n–k spindle graph as follows: take n copies of Pk and two extra vertices x and y, and then join the left endpoint (respectively, the right endpoint) of each Pk to x (respectively, y), the resulting graph is denoted by S(n, k), and called the n–k spindle graph. In this paper, we determine the optimal pebbling number for spindle graphs.

Constructions for the Optimal Pebbling of Grids

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.