Optimal pebbling of grids

Graph pebbling has its origin in number theory. It is a model for the transportation of resources. Starting with a pebble distribution on the vertices of a simple connected graph, a pebbling move removes two pebbles from a vertex and adds one pebble at an adjacent vertex. We can think of the pebbles as fuel containers. Then the loss of the pebble during a move is the cost of transportation. A vertex is called reachable if a pebble can be moved to that vertex using pebbling moves. There are several questions we can ask about pebbling. One of them is: How can we place the smallest number of pebbles such that every vertex is reachable (optimal pebbling number)? For a comprehensive list of references for the extensive literature see the survey papers [4, 5, 6]. Results on special grids can be found in [2] where the authors show that πopt(Pn P2) = πopt(Cn P2) = n apart from a few smaller case, and in [11] the author gave upper bounds for the optimal pebbling number of various grids. In the present paper we give better upper and lower bounds for the optimal pebbling numbers of large grids (Pn Pn). Graph rubbling is an extension of graph pebbling. In this version, we also allow a move that removes a pebble each from the vertices v and w that are adjacent to a vertex u, and adds a pebble at vertex u. The basic theory of rubbling and optimal rubbling is developed in [1]. The rubbling number of complete m-ary trees are studied in [3], while the rubbling number of caterpillars are determined in [10]. In [7] the authors gives upper and lower bounds for the rubbling number of diameter 2 graphs. In the present paper we determine the optimal rubbling number of ladders (Pn P2), prisms (Cn P2) and Möblus-ladders. We also give upper and lower bounds for the optimal rubbling numbers of large grids (Pn Pn).