Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph GG consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define ft(G)ft(G) as the smallest number such that whenever ft(G)ft(G) pebbles are on GG, we can move tt pebbles to any specified, but arbitrary vertex. Graham conjectured that f1(G×H)≤f1(G)f1(H)f1(G×H)≤f1(G)f1(H) for any connected GG and HH. We define the αα-pebbling number α(G)α(G) and prove that α(Cpj×⋯×Cp2×Cp1×G)≤α(Cpj)⋯α(Cp2)α(Cp1)α(G)α(Cpj×⋯×Cp2×Cp1×G)≤α(Cpj)⋯α(Cp2)α(Cp1)α(G) when none of the cycles is C5C5, and GG satisfies one more criterion. We also apply this result with G=C5×C5G=C5×C5 by showing that C5×C5C5×C5 satisfies Chung’s two-pebbling property, and establishing bounds for ft(C5×C5)ft(C5×C5).