Upper Bound on the Optimal Rubbling Number in graphs with given minimum degree

Abstract

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable. Czygrinow proved that the optimal pebbling number of a graph is at most $\frac{4n}{\delta+1}$, where $n$ is the number of the vertices and $\delta$ is the minimum degree of the graph. We improve this result and show that the optimal pebbling number is at most $\frac{3.75n}{\delta+1}$