On thet-pebbling number and the2t-pebbling property of graphs

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 and placing one on an adjacent vertex. The t-pebbling [email protected]"t(G) is the smallest positive integer such that, for every distribution of @p"t(G) pebbles and every vertex v, t pebbles can be moved to v. For t=1, Graham conjectured that @p"1([email protected]?H)@[email protected]"1(G)@p"1(H) for any connected graphs G and H, where [email protected]?H denotes the Cartesian product of G and H. Herscovici further conjectured that @p"s"t([email protected]?H)@[email protected]"s(G)@p"t(H). In this paper, we show that @p"s"t([email protected]?G)@[email protected]"s(T)@p"t(G), @p"s"t(K"[email protected]?G)@[email protected]"s(K"n)@p"t(G) and @p"s"t(C"2"[email protected]?G)@[email protected]"s(C"2"n)@p"t(G) when G has the 2t-pebbling property, T is a tree, K"n is the complete graph on n vertices, and C"2"n is the cycle on 2n vertices, which confirms a conjecture due to Lourdusamy. Moreover, we also show that any graph G with diameter 2 and @p"t(G)[email protected](G)+4(t-1) has the 2t-pebbling property, which extends a result of Pachter et al.