Constructions for the optimal pebbling of grids

In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics] the authors conjecture that if every vertex of an infinite square grid is reachable from a pebble distribution, then the covering ratio of this distribution is at most $3.25$. First we present such a distribution with covering ratio $3.5$, disproving the conjecture. The authors in the above paper also claim to prove that the covering ratio of any pebble distribution is at most $6.75$. The proof contains some errors. We present a few interesting pebble distributions that this proof does not seem to cover and highlight some other difficulties of this topic

Optimal pebbling of grids

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 ($P_n\square P_n$)

More on the 22-restricted optimal pebbling number

Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the weight of $f$ is $w(f)=\sum_{u\in V}f(u)$ which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one. A pebbling configuration $f$ is said to be solvable if for every vertex $v$, there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$. A pebbling configuration $f$ is a $t$-restricted pebbling configuration (abbreviated $t$RPC) if $f(v)\leq t$ for all $v\in V$. The $t$-restricted optimal pebbling number $\pi_t^*(G)$ is the minimum weight of a solvable $t$RPC on $G$. Chellali et.al. [Discrete Appl. Math. 221 (2017) 46-53] characterized connected graphs $G$ having small $2$-restricted optimal pebbling numbers and characterization of graphs $G$ with $\pi_2^*(G)=5$ stated as an open problem. In this paper, we solve this problem. We improve the upper bound of the $2$-restricted optimal pebbling number of trees of order $n$. Also, we study $2$-restricted optimal pebbling number of some grid graphs, corona and neighborhood corona of two specific graphs.Comment: 12 pages, 11 figure

The optimal pebbling number of staircase graphs

Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can be placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs. © 2018 Elsevier B.V

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