Pebbling on Graph Products and other Binary Graph Constructions

Pebbling on graphs is a two-player game which involves repeatedly moving a pebble from one vertex to another by removing another pebble from the first vertex. The pebbling number $\pi(G)$ is the least number of pebbles required so that, regardless of the initial configuration of pebbles, a pebble can reach any vertex. Graham conjectured that the pebbling number for the cartesian product, $G \hspace{1mm}\square\hspace{1mm} H$, is bounded above by $\pi(G) \pi(H)$. We show that $\pi(G\hspace{1mm}\square\hspace{1mm} H) \le 2\pi(G) \pi(H)$ and, more sharply, that $\pi(G \hspace{1mm}\square\hspace{1mm} H) \le (\pi(G)+|G|) \pi(H)$. Furthermore, we provide similar results for other graph products and graph operations.