Optimal pebbling of grids

Katona, Gyula Y. and Győri, Ervin and Papp, László F. (2015) Optimal pebbling of grids. In: 13th Cologne-Twente Workshop on Graphs & Combinatorial Optimization, May 26-28, 2015, Isztanbul.

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Abstract

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices $v$ and $w$ adjacent to a vertex $u$, and an extra pebble is added at vertex $u$. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The optimal pebbling (rubbling) number is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable using pebbling (rubbling) moves. We determine the optimal rubbling number of ladders ($P_n\square P_2$), prisms ($C_n\square P_2$) and M\"oblus-ladders. We also give upper and lower bounds for the optimal pebbling and rubbling numbers of large grids ($P_n\square P_n$).

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