Critical Pebbling Numbers of Graphs

We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the graph, this yields 7 graph parameters. We determine the relationships between these parameters. We investigate properties of the $r$-critical pebbling number, and distinguish between greedy graphs, thrifty graphs, and graphs for which the $r$-critical pebbling number is $2^d$.Comment: 26 page

Optimal Pebbling in Products of Graphs

We prove a generalization of Graham's Conjecture for optimal pebbling with arbitrary sets of target distributions. We provide bounds on optimal pebbling numbers of products of complete graphs and explicitly find optimal $t$-pebbling numbers for specific such products. We obtain bounds on optimal pebbling numbers of powers of the cycle $C_5$. Finally, we present explicit distributions which provide asymptotic bounds on optimal pebbling numbers of hypercubes.Comment: 28 pages, 1 figur

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