following families of graphs: , where is a complete graph on n vertices. , where is a path graph on n vertices. , where is a wheel graph on n vertices. Graham's pebbling conjecture credited Ronald Graham with the conjecture that the pebbling number of a Cartesian product of graphs is at most equal to the product of the pebbling numbers of the factors in the product. This has come to be known as Graham's pebbling conjecture. , it remains unsolved, although special cases are known. γ(G) — the cover pebbling number of a graph Crull et al. introduced the concept of cover pebbling. γ(G), the cover pebbling number of a graph is the minimum number of pebbles needed so that from any initial arrangement of the pebbles, after a series of pebbling moves, the graph is covered: there is at least one pebble on every vertex. A result called the stacking theorem finds the cover pebbling number for any graph. The stacking theorem According to the stacking theorem, the initial configuration of pebbles that requires the most pebbles to be cover solved happens when all pebbles are placed on a single vertex. Based on this observation, define for every vertex v in G, where d(u, v) denotes the distance from u to v. Then the cover pebbling number is the largest s(v) that results. γ(G) for families of graphs The cover pebbling number is known for the following families of graphs: , where is a complete graph on n vertices. , where is a path on n vertices. , where is a wheel graph on n vertices. See also Pebble game Proof of space References Further reading Graph invariants