Pebbling numbers of the Cartesian product of cycles and graphs

The pebbling number f(G) of a graphG is the least p such that, no matter how 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 , we have f(G × H) ≤ f(G)f(H). If the graph G satisfies the odd 2-pebbling property, we will prove that f(C4k+3 ×G) ≤ f(C4k+3)f(G) and f(M(C2n)×G) ≤ f(M(C2n))f(G), where C4k+3 is the odd cycle of order 4k + 3 and M(C2n) is the middle graph of the even cycle C2n.