Optimal pebbling number of graphs with given minimum degree

Consider a distribution of pebbles on a connected graph G. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number π ^∗ (G) is the smallest number of pebbles that we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most [Formula presented], where δ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If diam(G)≥3 then we further improve the bound to π ^∗ (G)≤[Formula presented]. On the other hand, we show that, for arbitrary large diameter and any ϵ>0, there are infinitely many graphs whose optimal pebbling number is bigger than [Formula presented]−ϵ[Formula presented].