Lourdusamy’s conjecture on ZZ_n(C_2k) × G

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f (G) of a connected graph G is the smallest positive integer such that every distribution of f (G) pebbles on the vertices of G, we can move a pebble to any target vertex. The t-pebbling number f_t (G) of a connected graph G is the smallest positive integer such that every distribution of f_t (G) pebbles on the vertices of G, we can move t pebbles to any target vertex by a sequence of pebbling moves. Graham conjectured that for any connected graph G and H, f (G × H) ≤ f (G) f (H). Lourdusamy further conjectured that f_t (G × H) ≤ f (G) f_t (H) for any positive integer t. In this paper, we show that Lourdusamy’s Conjecture is true when G is a zig-zag chain graph of n copies of even cycles and H is a graph having 2t- pebbling property.