Stacking and clearing in graph pebbling

TL;DR

This paper introduces and analyzes two new graph parameters, stack(G) and clear(G), related to pebbling configurations, providing bounds, exact values for specific graphs, and conjectures for trees and cycles.

Contribution

The paper defines stack(G) and clear(G), establishes their properties, bounds, exact values for certain graphs, and proposes conjectures for cycles and trees.

Findings

stack(G) and clear(G) are well-defined for connected graphs

Exact values for complete, bipartite, and path graphs

Upper bounds involving graph diameter and size

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.