Pebbling and Graham's conjecture (Q1841936)

Given an arrangement of pebbles on the vertices of a graph \(G\), a pebbling move is defined to consist of removing two pebbles from one vertex and placing one pebble on an adjacent vertex [see, i.e., \textit{F. R. K. Chung}, SIAM J. Discrete Math. 2, No. 4, 467-472 (1989; Zbl 0727.05043)]. Let \(f(G,v)\) be the minimum integer which guarantees that any starting pebble configuration allows reaching a configuration with at least one pebble on \(v\) through repeating pebbling moves. Let the pebbling number of \(G\) be the maximum of \(f(G,v)\) for all vertices \(v\). The author proves that the conjecture \(f(G\times H)\leq f(G)f(H)\) holds if \(H\) is a complete multipartite graph and \(G\) has Chung's 2-pebbling property. He also gives an infinite class of graphs which are not 2-pebbling.