More on graph pebbling number

Abstract

Let $G=(V,E)$ be a simple graph. A function $\phi:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\sum_{u\in V}\phi(u)$ is called the size of $\phi$ 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 $\phi(u)$ by two and increases $\phi(v)$ by one.  Given a specified target vertex $r$ we say that $\phi$ 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 $\phi$ 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 $\pi_t(G,r)$. The $t$-fold pebbling number of $G$ is defined to be $\pi_t(G)= max_{r\in V(G)}\pi_t(G,r)$. When $t=1$, we simply write $\pi(G)$, 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.