Pebbling and Optimal Pebbling in Graphs
  Given a distribution of pebbles on the vertices of a graph G, a {\it pebbling move} takes two pebbles from one vertex and puts one on a neighboring vertex. The {\it pebbling number} \Pi(G) is the minimum k such that for every distribution of k pebbles and every vertex r, it is possible to move a pebble to r. The {\it optimal pebbling number} \Pi_{OPT}(G) is the minimum k such that some distribution of k pebbles permits reaching each vertex.   We give short proofs of prior results on these parameters for paths, cycles, trees, and hypercubes, a new linear-time algorithm for computing \Pi(G) on trees, and new results on \Pi_{OPT}(G). If G is a connected n-vertex graph, then \Pi_{OPT}(G)\le\ceiling{2n/3}, with equality for paths and cycles. If \bf{G} is the family of n-vertex connected graphs with minimum degree k, then 2.4\le \max_{G\in \bf{G}} \Pi_{OPT}(G) \frac{k+1}{n}\le 4 when k>15 and k is a multiple of 3. Finally, \Pi_{OPT}(G)\le 4^tn/((k-1)^t+4^t) when G is a connected n-vertex graph with minimum degree k and girth at least 2t+1. For t=2, a more precise version of this last bound is \Pi_{OPT}(G)\le 16n/(k^2+17). 
D. P. Bunde, E. W. Chambers, D. Cranston, K. Milans, and D. B. West