A new lower bound on the pebbling number of the grid

TL;DR

This paper improves the lower bound on the minimum number of pebbles needed to reach every vertex in an n by m grid graph, advancing understanding of pebbling numbers in grid structures.

Contribution

The authors establish a tighter lower bound on the optimal pebbling number for grid graphs, enhancing previous bounds by a significant margin.

Findings

New lower bound of approximately 0.1781nm for the pebbling number

Improved theoretical understanding of pebbling in grid graphs

Refined bounds contribute to graph pebbling theory

Abstract

A pebbling move on a graph consists of removing $2$ pebbles from a vertex and adding $1$ pebble to one of the neighbouring vertices. A vertex is called reachable if we can put $1$ pebble on it after a sequence of moves. The optimal pebbling number of a graph is the minimum number $m$ such that there exists a distribution of $m$ pebbles so that each vertex is reachable. For the case of a square grid $n \times m$, Gy\H{o}ri, Katona and Papp recently showed that its optimal pebbling number is at least $\frac{2}{13}nm \approx 0.1538nm$ and at most $\frac{2}{7}nm +O(n+m) \approx 0.2857nm$. We improve the lower bound to $\frac{5092}{28593}nm +O(m+n) \approx 0.1781nm$.