Lourdusamy's conjecture for Kn-k1 × Km-k2

Abstract

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of removing two pebbles from one vertex and placing one on an adjacent vertex. The t-pebbling number f_t(G) of a simple 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. Denote f₁(G) =/(G). Graham conjectured that for any connected graphs 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 integers t. In this paper, we show that Lourdusamy's conjecture is true when G = K_n^-k, a graph obtained from K_n by deleting k independent edges, and H is a graph having the 2t-pebbling property.