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 guaran...

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Veröffentlicht in:Graphs and combinatorics 2020-05, Vol.36 (3), p.803-829
Hauptverfasser: Győri, Ervin, Katona, Gyula Y., Papp, László F.
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Katona, Gyula Y.
Papp, László F.
description 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 □ P m 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 π opt . We apply this technique to prove that π opt ( P n □ P m ) ≥ 2 13 n m . Our method also gives a new proof for π opt ( P n ) = π opt ( C n ) = 2 n 3 .
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subjects Combinatorics
Engineering Design
Graph theory
Lower bounds
Mathematics
Mathematics and Statistics
Original Paper
title Optimal Pebbling Number of the Square Grid
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