A note on graph pebbling

We say that a graph G is Class 0 if its pebbling number is exactly equal to its number of vertices. For a positive integer d, let k(d) denote the least positive integer so that every graph G with diameter at most d and connectivity at least k(d) is Class 0. The existence of the function k was conjectured by Clarke, Hochberg and Hurlbert, who showed that if the function k exists, then it must satisfy k(d) = 0(2^d/d). In this note, we show that k exists and satisfies k(d) = O(2^2d). We also apply this result to improve the upper bound on the random graph threshold of the Class 0 property.