Optimal pebbling number of the square grid

TL;DR

This paper introduces a new method to establish lower bounds on the optimal pebbling number of square grid graphs, improving understanding of pebbling configurations and providing new proofs for known cases.

Contribution

The paper presents a novel technique for bounding the optimal pebbling number and applies it to square grid graphs, offering improved bounds and new proofs for specific graph classes.

Findings

Proved that the optimal pebbling number of $P_n\square P_m$ is at least (2/13)nm.

Provided a new proof that $\pi_{opt}(P_n)=\pi_{opt}(C_n)=\lceil 2n/3 \rceil$.

Introduced a new method for lower bounds on pebbling numbers.

Abstract

A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $\pi_{opt}$ is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph $P_n\square P_m$ was investigated in several papers. In this paper, we present a new method using some recent ideas to give a lower bound on $\pi_{opt}$. We apply this technique to prove that $\pi_{opt}(P_n\square P_m)\geq \frac{2}{13}nm$. Our method also gives a new proof for $\pi_{opt}(P_n)=\pi_{opt}(C_n)=\left\lceil\frac{2n}{3}\right\rceil$.