Stacking and clearing in graph pebbling
Tam\'as Csern\'ak, Lajos Soukup

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 +…
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