The [formula omitted]-pebbling number of [formula omitted

Abstract

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 numberπt(G) of a connected graph G is the smallest positive integer such that from every distribution of πt(G) pebbles on G, t pebbles can be moved to any specified target vertex of G. For t=1, Graham conjectured that π1(G□H)≤π1(G)π1(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 C5×C5, Discrete Math. 187 (1998) 123–135] proved that π1(C5□C5)=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 πt(C5□C5)=16t+7. In this paper, we confirm this conjecture.