On the tt-pebbling number and the 2t2t-pebbling property of graphs

Given a distribution of pebbles on the vertices of a connected graph GG, a pebbling move on GG consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The tt-pebbling number  πt(G)πt(G) is the smallest positive integer such that, for every distribution of πt(G)πt(G) pebbles and every vertex vv, tt pebbles can be moved to vv. For t=1t=1, Graham conjectured that π1(G□H)≤π1(G)π1(H) for any connected graphs GG and HH, where G□H denotes the Cartesian product of GG and HH. Herscovici further conjectured that πst(G□H)≤πs(G)πt(H). In this paper, we show that πst(T□G)≤πs(T)πt(G), πst(Kn□G)≤πs(Kn)πt(G) and πst(C2n□G)≤πs(C2n)πt(G) when GG has the 2t2t-pebbling property, TT is a tree, KnKn is the complete graph on nn vertices, and C2nC2n is the cycle on 2n2n vertices, which confirms a conjecture due to Lourdusamy. Moreover, we also show that any graph GG with diameter 2 and πt(G)=π(G)+4(t−1)πt(G)=π(G)+4(t−1) has the 2t2t-pebbling property, which extends a result of Pachter et al.