Optimal t-pebbling on paths and cycles

A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbours. For t≥1, the optimal t-pebbling number of a graph G, f′t(G), is the minimum number of pebbles necessary so that from some initial distribution of them it is possible to move t pebbles to any target vertex by a sequence of pebbling moves. f′(G)=f′1(G) be the optimal pebbling number of G. Here the optimal t-pebbling numbers of the path Pn and the cycle C5 were given, respectively. In the final section, it was obtained that f′9t(P2×P3)=20t, f′9t+1(P2×P3)=20t+3, and 20t+2r+1≤f′9t+r(P2×P3)≤20t+2r+2, for 2≤r≤8, the last equality holds for r=5,6,7,8.