The Optimal Rubbling Number of Paths, Cycles, and Grids

Abstract

A pebbling move on a graph G consists of the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed, which is also called the strict rubbling move. In this new move, one pebble each is removed from u and v adjacent to a vertex w, and one pebble is added on w. The rubbling number of a graph G is the smallest number m, such that one pebble can be moved to each vertex from every distribution with m pebbles. The optimal rubbling number of a graph G is the smallest number m, such that one pebble can be moved to each vertex from some distribution with m pebbles. In this paper, we give short proofs to determine the rubbling number of cycles and the optimal rubbling number of paths, cycles, and the grid P2×Pn; moreover, we give an upper bound of the optimal rubbling number of Pm×Pn.