Menu
Given a distribution of pebbles on the vertices of a connected graph \( G \), a pebbling move on \( G \) consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The \( t \)-pebbling number \( \pi_t(G) \) is the smallest positive integer such that for every distribution of \( \pi_t(G) \) pebbles and every vertex \( v \), \( t \) pebbles can be moved to \( v \). For \( t = 1 \), Graham conjectured that \( \pi_1(G \Box H) \leq \pi_1(G)\pi_1(H) \) for any connected graphs \( G \) and \( H \), where \( G \Box H \) denotes the Cartesian product of \( G \) and \( H \). Herscovici further conjectured that \( \pi_{st}(G \Box H) \leq \pi_s(G)\pi_t(H) \) for any positive integers \( s \) and \( t \). Lourdusamy [A. Lourdusamy, “\(t\)-pebbling the product of graphs”, Acta Ciencia Indica, XXXII(1)(2006), 171-176] also conjectured that \( \pi_t(C_m \Box C_n) \leq \pi_1(C_m)\pi_t(C_n) \) for cycles \( C_m \) and \( C_n \). In this paper, we show that \( \pi_{st}(C_m \Box C_n) \leq \pi_s(C_m)\pi_t(C_n) \), which confirms this conjecture due to Lourdusamy.