Covering cover pebbling number of products of paths

Abstract

There are a variety of pebbling numbers, such as classical pebbling number, cover pebbling number, and covering cover pebbling number. In this paper we determine the covering cover pebbling number for Cartesian products of paths. The covering cover pebbling number of a graph, G, is the smallest number of pebbles, n, required such that any distribution of n pebbles onto the vertices of G can be, through a sequence of pebbling moves, redistributed so that C, a vertex cover of G, is pebbled. Traditionally, a pebbling move is defined as the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. In this paper we provide an alternative proof for the covering cover pebbling number of cycles and prove the covering cover pebbling number for a Cartesian product of paths.