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 $P_2\times P_n$ ; moreover, we give an upper bound of the optimal rubbling number of $P_m\times P_n$ .