The concept of graph pebbling was introduced by Lagarias and Saks as a way to prove a number-theoretic conjecture of Erdos and Lemke. Chung successfully carried out their method by proving that the pebbling number of the $n$-cube is equal to the number of its vertices. Then Graham conjectured that $f(G\Box H)\le f(G)f(H)$, which would generalize her result. We will survey partial results on this question as well as introduce new results and conjectures about the behavior of $f$.
For example, by characterizing which diameter two graphs $G$ have $f(G)=n(G)$, we prove that every 3-connected diameter two graph $G$ has $f(G)=n(G)$, extending results of Pachter, Snevily and Voxman. We also use this characterization to show the same holds for almost all graphs. We conjecture that there is a function $k(d)$ such that if $G$ is a graph with $diam(G)=d$ and $\kappa (G)\ge k(d)$ then $f(G)=n(G)$. We prove that if $k$ exists then $k(d)\ge 2^d/d$. We believe that $k(d)\le 2^d$.
If time permits we will discuss some of the many open problems involving topics like greedy pebbling, tree-solvability, $p$-pebbling, threshold functions and the 2-pebbling property. Much of this is joint work with Tom Clarke and Rob Hochberg.