Pebbling on $C_{4k+3}\times G$ and $M(C_{2n})\times G$

Abstract

The pebbling number of a graph $G$, $f(G)$, is the least $p$ such that, however $p$ pebbles are placed on the vertices of $G$, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs $G$ and $H$, $f(G\times H)\le f(G)f(H)$. If the graph $G$ satisfies the odd two-pebbling property, we will prove that $f(C_{4k+3}\times G)\le f(C_{4k+3})f(G)$ and $f(M(C_{2n})\times G)\le f(M(C_{2n}))f(G)$, where $C_{4k+3}$ is the odd cycle of order $4k+3$ and $M(C_{2n})$ is the middle graph of the even cycle $C_{2n}$.