Herscovici’s Conjecture on the Product of the Thorn Graphs of the Complete Graphs

Abstract

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 \(f_t(G)\) of a simple connected graph \(G\) is the smallest positive integer such that for 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 graphs G and \(H\), \(f_1(G\times H)\leqslant f_1(G)f_1(H)\). Herscovici further conjectured that \(f_{st}(G\times H)\leqslant f_s(G)f_t(H)\) for any positive integers \(s\) and \(t\). Wang et al. (Discret Math, 309: 3431–3435, 2009) proved that Graham’s conjecture holds when \(G\) is a thorn graph of a complete graph and \(H\) is a graph having the \(2\)-pebbling property. In this paper, we further show that Herscovici’s conjecture is true when \(G\) is a thorn graph of a complete graph and \(H\) is a graph having the \(2t\)-pebbling property.