Optimal pebbling number of the square grid

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 $π_{opt}$ 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 $π_{opt}$. We apply this technique to prove that $π_{opt}(P_n\square P_m)\geq \frac{2}{13}nm$. Our method also gives a new proof for $π_{opt}(P_n)=π_{opt}(C_n)=\left\lceil\frac{2n}{3}\right\rceil$.

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