  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)=\Omega(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.
