Conditions for Weighted Cover Pebbling of Graphs

In a graph G with a distribution of pebbles on its vertices, a pebbling move is the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. A weight function on G is a non-negative integer-valued function on the vertices of G. A distribution of pebbles on G covers a weight function if there exists a sequence of pebbling moves that gives a new distribution in which every vertex has at least as many pebbles as its weight. In this paper we give some necessary and some sufficient conditions for a distribution of pebbles to cover a given weight function on a connected graph G. As a corollary, we give a simple formulation for the `weighted cover pebbling number' of a weight function W and a connected graph G, defined by Crull et al. to be the smallest number m such that any distribution on G of m pebbles is a cover for W. Also, we prove a cover pebbling variant of Graham's Conjecture for pebbling.