Pebbling Number of Bi-Wheel: A Diameter Three Class $0$ Graph

Abstract

Given a configuration of pebbles on the vertices of a graph $G$, a pebbling move consists of taking two pebbles off a vertex $v$ and putting one of them back on a vertex adjacent to $v$. A graph is called $pebbleable$ if for each vertex $v$ there is a sequence of pebbling moves that would place at least one pebble on $v$. The $pebblingnumber$ of a graph $G$, is the smallest integer $m$ such that $G$ is pebbleable for every configuration of $m$ pebbles on $G$. A graph $G$ is said to be class $0$ if the pebbling number of $G$ is equal to the number of vertices in $G$. We prove that $Bi-wheels$, a class of diameter three graphs, are class $0$.