More on graph pebbling number
Saeid Alikhani, Fatemeh Aghaei
Abstract
Let $G=(V,E)$ be a simple graph. A function $φ:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\sum_{u\in V}φ(u)$ is called the size of $φ$ 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 $φ(u)$ by two and increases $φ(v)$ by one. Given a specified target vertex $r$ we say that $φ$ is $t$-fold $r$-solvable, if some sequence of pebbling steps places at least $t$ pebbles on $r$. Conversely, if no such steps exist, then $φ$ is $r$-unsolvable. The minimum positive integer $m$ such that every configuration of size $m$ on the vertices of $G$ is $t$-fold $r$-solvable is denoted by $π_t(G,r)$. The $t$-fold pebbling number of $G$ is defined to be $π_t(G)= max_{r\in V(G)}π_t(G,r)$. When $t=1$, we simply write $π(G)$, which is the pebbling number of $G$. In this note, we study the pebbling number for some specific graphs. Also we investigate the pebbling number of corona and neighbourhood corona of two graphs.