The proof of a conjecture due to Snevily

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 the2-pebbling property if, for any distribution with more than 2 f ( 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. A graph G without the 2-pebbling property is called a Lemke graph. Snevily and Foster [H.S. Snevily, J.A. Foster, The 2-pebbling property and a conjecture of Graham’s, Graphs and Combin. 16 (2000), 231–244] defined an infinite family { L 1 , L 2 , … } of possible Lemke graphs, and conjectured that L k is a Lemke graph for each k . In this paper, we prove this conjecture.