Optimal Pebbling Number of the Square Grid

Abstract

AbstractA 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 $$\pi _{{{\,\mathrm{opt}\,}}}$$πopt 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 $$P_n\square P_m$$Pn□Pm was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on $$\pi _{{{\,\mathrm{opt}\,}}}$$πopt. We apply this technique to prove that $$\pi _{{{\,\mathrm{opt}\,}}}(P_n\square P_m)\ge \frac{2}{13}nm$$πopt(Pn□Pm)≥213nm. Our method also gives a new proof for $$\pi _{{{\,\mathrm{opt}\,}}}(P_n)=\pi _{{{\,\mathrm{opt}\,}}}(C_n)=\left\lceil \frac{2n}{3}\right\rceil$$πopt(Pn)=πopt(Cn)=2n3.