The 2-pebbling Property for Dense Graphs

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 pebbling number f(G) is the smallest number m such that for every distribution of m pebbles and every vertex v, a pebble can be moved to v. A graph G is said to have the 2-pebbling property if for any distribution with more than 2f(G) - q pebbles, where q is the number of vertices with at least one pebble, it is possible, using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G(s, t) has the 2-pebbling property, where G(s, t) is a bipartite graph with partite sets of size s and t (s >= t). Similarly, the l-pebbling number f(l)(G) is the smallest number m such that for every distribution of m pebbles and every vertex v, l pebbles can be moved to v. Herscovici et al. conjectured that f(l)(G) <= 1.5n + 8l - 6 for the graph G with diameter 3, where n = vertical bar V(G)vertical bar. In this paper, we prove that if s >= 15 and G(s, t) has minimum degree at least [s+1/2], then f(G(s, t)) = s + t, G(s, t) has the 2-pebbling property and f(l)(G(s, t)) <= s + t + 8(l- 1). In other words, we extend a result due to Czygrinow and Hurlbert, and show that the above Snevily conjecture and Herscovici et al. conjecture are true for G(s, t) with s >= 15 and minimum degree at least [s+1/2].