The -pebbling number of

Given a distribution of pebbles on the vertices of a connected graph G , a pebbling move on G consists of taking two pebbles off one vertex, throwing one away, and putting the other pebble on an adjacent vertex. The t-pebbling number π t ( G ) of a connected graph G is the smallest positive integer such that from every distribution of π t ( G ) pebbles on G , t pebbles can be moved to any specified target vertex of G . For t = 1 , Graham conjectured that π 1 ( G □ H ) ≤ π 1 ( G ) π 1 ( H ) for any connected graphs G and H , where G □ H denotes the Cartesian product of G and H . Herscovici and Higgins [D.S. Herscovici, A.W. Higgins, The pebbling number of C 5 × C 5 , Discrete Math. 187 (1998) 123–135] proved that π 1 ( C 5 □ C 5 ) = 25 . Herscovici [D.S. Herscovici, Graham’s pebbling conjecture on products of many cycles, Discrete Math. 308 (2008) 6501–6512] conjectured that if t ≥ 2 , then π t ( C 5 □ C 5 ) = 16 t + 7 . In this paper, we confirm this conjecture.