Abstract.
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)=Ω(2d/d). In this note, we show that k exists and satisfies k(d)=O(22d). We also apply this result to improve the upper bound on the random graph threshold of the Class 0 property.
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Received: April 19, 1999 Final version received: February 15, 2000
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Czygrinow, A., Hurlbert, G., Kierstead, H. et al. A Note on Graph Pebbling. Graphs Comb 18, 219–225 (2002). https://doi.org/10.1007/s003730200015
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DOI: https://doi.org/10.1007/s003730200015