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 × P n ; moreover, we give an upper bound of the optimal rubbling number of P m × P n .

Monophonic rubbling number of some standard graphs

“…Nanbor Seiben briefly discussed graph pebbling number and graph rubbling number in [7] . Zheng Jiang Xia et al computed the rubbling number on the cycle graph and optimal rubbling number of paths, cycles, and grid on graphs and upper bound of the graph in [8] . Laszlo F. Papp et al computed the lower bounds on both the optimal pebbling and rubbling number using the distance k domination number in [9] .…”