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Optimal Pebbling on Grids

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Abstract

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each distribution of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We say such a distribution is solvable. The optimal pebbling number of G, denoted \(\varPi _{OPT}(G)\), is the least k such that some particular distribution of k pebbles is solvable. In this paper, we strengthen a result of Bunde et al. relating to the optimal pebbling number of the 2 by n square grid by describing all possible optimal configurations. We find the optimal pebbling number for the 3 by n grid and related structures. Finally, we give a bound for the analogue of this question for the infinite square grid.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Davidson College, Davidson, NC, 28035, USA

    Chenxiao Xue & Carl Yerger

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  1. Chenxiao Xue
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  2. Carl Yerger
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Correspondence to Carl Yerger.

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Xue, C., Yerger, C. Optimal Pebbling on Grids. Graphs and Combinatorics 32, 1229–1247 (2016). https://doi.org/10.1007/s00373-015-1615-5

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  • DOI: https://doi.org/10.1007/s00373-015-1615-5

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