The $t$-Pebbling Conjecture on Products of Complete $r$-Partite Graphs

Abstract

The $t$-pebbling number $f_t(G)$ of a graph $G$ is the least positive integer $m$ such that however these $m$ pebbles are placed on the vertices of $G$, we can move $t$ pebbles to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we study the generalized Graham’s pebbling conjecture $f_t(G\times H)\le f(G)f_t(H)$ for the product of graphs when $G$ is a complete $r$-partite graph and $H$ has a $2t$-pebbling property.