Pebbling and optimal pebbling in graphs

Given a distribution of pebbles on the vertices of a graph G, a pebbling move takes two pebbles from one vertex and puts one on a neighboring vertex. The pebbling number (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 optimal pebbling number 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 (G) on trees, and new results on OPT (G). If G is a connected n-vertex graph, then OPT (G) 2n/3 , with equality for paths and cycles. If G is the family of n-vertex connected graphs with minimum degree k, then 2.4 maxGG OPT (G) k+1 n 4 when k > 15 and k is a multiple of 3. Finally, OPT (G) 4tn/((k - 1)t + 4t) when G is a connected

• Type: Journal
• Year: 2008
• Where: JGT
• Authors: David P. Bunde, Erin W. Chambers, Daniel W. Cranston, Kevin Milans, Douglas B. West
• Keywords: JGT 2008, Neighboring Vertex, Pebbles, Pebbling Number