On the [formula omitted]-pebbling number and the [formula omitted]-pebbling property of graphs

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 and placing one on an adjacent vertex. The t-pebbling numberπt(G) is the smallest positive integer such that, for every distribution of πt(G) pebbles and every vertex v, t pebbles can be moved to v. 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 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 G has the 2t-pebbling property, T is a tree, Kn is the complete graph on n vertices, and C2n 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 πt(G)=π(G)+4(t−1) has the 2t-pebbling property, which extends a result of Pachter et al.