Almost-all graphs greedy and tree-solvable conjecture

A graph is greedy if every configuration of $\pi(G)$ pebbles can be solved at any specified root using only greedy pebbling steps. It is tree-solvable if every configuration of size $\pi(G)$ can be solved so that the edges used by pebbling steps form an acyclic graph. Almost-all graphs conjecture. Almost every graph is greedy and tree-solvable.

The paper gives examples of graphs that are neither greedy nor tree-solvable, and notes that even semi-greediness need not be preserved under graph products; the asymptotic assertion itself is left unresolved.