Lourdusamy’s conjecture on ZZn(C2k) × G

Abstract

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 ft (G) of a connected graph G is the smallest positive integer such that every distribution of ft (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 ft (G × H) ≤ f (G) ft (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.