Thresholds for Pebbling on Grids
Neal Bushaw, Nathan Kettle

TL;DR
This paper determines the probabilistic thresholds for pebbling configurations on grid graphs and paths, introduces a new centrality concept and weight lemma, and discusses related conjectures and prior work.
Contribution
It establishes the weak and strong thresholds for q-pebbling on grids and paths, introduces a novel centrality notion and weight lemma, and extends previous results to the probabilistic setting.
Findings
Determined the weak threshold for q-pebbling on grids as size increases.
Established the strong threshold for q-pebbling on paths.
Introduced a new centrality concept and a pebbling weight lemma.
Abstract
Given a connected graph and a configuration of pebbles on the vertices of G, a -pebbling step consists of removing pebbles from a vertex, and adding a single pebble to one of its neighbors. Given a vector , -pebbling consists of allowing -pebbling in coordinate . A distribution of pebbles is called solvable if it is possible to transfer at least one pebble to any specified vertex of via a finite sequence of pebbling steps. In this paper, we determine the weak threshold for -pebbling on the sequence of grids for fixed and , as . Further, we determine the strong threshold for -pebbling on the sequence of paths of increasing length. A fundamental tool in these proofs is a new notion of centrality, and a sufficient condition for solvability based on the well used pebbling weight…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Mathematical Dynamics and Fractals · Geometric and Algebraic Topology
