On the t t -pebbling number and the 2t 2 t -pebbling property of graphs ?
Given a distribution of pebbles on the vertices of a connected graph G G , a pebbling move on G G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The t t -pebbling number ?t(G) ? t ( G ) is the smallest positive integer such that, for every distribution of ?t(G) ? t ( G ) pebbles and every vertex v v , t t pebbles can be moved to v v . For t=1 t = 1 , Graham conjectured that ? 1 ( G ? H ) ? ? 1 ( G ) ? 1 ( H ) for any connected graphs G G and H H , where G ? H denotes the Cartesian product of G G and H H . Herscovici further conjectured that ? s t ( G ? H ) ? ? s ( G ) ? t ( H ) . In this paper, we show that ? s t ( T ? G ) ? ? s ( T ) ? t ( G ) , ? s t ( K n ? G ) ? ? s ( K n ) ? t ( G ) and ? s t ( C 2 n ? G ) ? ? s ( C 2 n ) ? t ( G ) when G G has the 2t 2 t -pebbling property, T T is a tree, Kn K n is the complete graph on n n vertices, and C2n C 2 n is the cycle on 2n 2 n vertices, which confirms a conjecture due to Lourdusamy. Moreover, we also show that any graph G G with diameter2 and ?t(G)=?(G)+4(t?1) ? t ( G ) = ? ( G ) + 4 ( t ? 1 ) has the 2t 2 t -pebbling property, which extends a result of Pachter et al.
pebbling grahams conjecture 2t 2 t -pebbling property 2t 2 t 2t 2 t 2 t