Pebbling on Graph Products and other Binary Graph Constructions

Abstract

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, $G\square H$, is bounded above by $\pi (G)\pi (H)$. We show that $\pi (G\square H)\le 2\pi (G)\pi (H)$ and, more sharply, that $\pi (G\square H)\le (\pi (G)+|G|)\pi (H)$. Furthermore, we provide similar results for other graph products and graph operations.

20 pages, 5 figures

Cited by in corpus (1)

• Computing Bounds on Product-Graph Pebbling Numbers