Lemke graphs and Graham's pebbling conjecture

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 π(G) is the smallest positive integer such that for every distribution of π(G) pebbles and every vertex v, a pebble can be moved to v. Graham conjectured that π(G□H)≤π(G)π(H) for any connected graphs G and H, where G□H denotes the Cartesian product of G and H. A graph G has the 2-pebbling property (respectively, the odd-2-pebbling property) if for any distribution with more than 2π(G)−q (respectively, 2π(G)−r) pebbles, where q is the number of vertices with at least one pebble (respectively, r is the number of vertices with an odd number of pebbles), it is possible, using pebbling moves, to get two pebbles to any vertex. Obviously, graphs which have the 2-pebbling property also have the odd-2-pebbling property. A graph G without the 2-pebbling property is called a Lemke graph. In this paper, we further extend the concept of the odd-2-pebbling property to the weak odd-2-pebbling property and show that all Lemke graphs found to date have the weak odd-2-pebbling property, but not the odd-2-pebbling property. Moreover, we also prove that π(L□T)≤π(L)π(T) and π(L□Kn)≤π(L)π(Kn), where T is a tree, Kn is the complete graph on n vertices and L is a known Lemke graph. The two evidences supporting Graham's conjecture differ from the previous evidences requiring that G and H have the 2-pebbling property.