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 kðdÞ denote the least positive integer so that every graph G with diameter at most d and connectivity at least kðdÞ 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 kðdÞ 1⁄4 Xð2d=dÞ. In this note, we show that k exists and satisfies kðdÞ 1⁄4 Oð22dÞ. We also apply this result to improve the upper bound on the random graph threshold of the Class 0 property.