Stacking and clearing in graph pebbling

Abstract

Suppose that pebbles are distributed on the vertices of a graph G. A pebbling step along an edge uv removes two pebbles from u and places one pebble on v. We introduce two new graph parameters: stack(G): the least integer t such that every configuration with t pebbles can be transformed, by a finite sequence of pebbling steps, into a configuration with all pebbles on a single vertex. clear(G): defined analogously, but requiring that from every configuration with t pebbles, all but one pebble can be removed. We prove that stack(G) is defined exactly for connected graphs, and that clear(G) is defined exactly for connected non-bipartite graphs. We also establish general upper bounds for these parameters; in particular, stack(G), clear(G) <= 2 |V(G)| 2^diam(G), where diam(G) denotes the diameter of G. Among our exact results are the equalities stack(K_n) = clear(K_n) = n + 1, stack(K_{m,n}) = 3 max{m,n} + 1, stack(P_n) = 2^n - 1. We also establish general lower bounds in terms of the independence number and odd closed walks. For cycles, the situation is more delicate. We prove the lower bounds stack(C_{2n}) >= 2^{n+1} - 1, clear(C_{2n+1}) >= 3 * 2^n - 2, and formulate the Almost Stacked Hypothesis, motivated by Sjostrand's cover pebbling theorem. Assuming this hypothesis, we obtain stack(C_{2n}) = 2^{n+1} - 1, clear(C_{2n+1}) = 3 * 2^n - 2. At present, we do not have a conjecture for the exact value of stack(C_{2n+1}). Finally, computational evidence leads us to a conjectural closed formula for the stacking number of a tree in terms of the distances and degrees of the vertices relative to a chosen root.