### 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) = 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.

Original language | English (US) |
---|---|

Pages (from-to) | 219-225 |

Number of pages | 7 |

Journal | Graphs and Combinatorics |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - 2002 |

### Fingerprint

### Keywords

- Connectivity
- Pebbling
- Threshold

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*18*(2), 219-225. https://doi.org/10.1007/s003730200015

**A note on graph pebbling.** / Czygrinow, Andrzej; Hurlbert, Glenn; Kierstead, Henry; Trotter, William T.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 18, no. 2, pp. 219-225. https://doi.org/10.1007/s003730200015

}

TY - JOUR

T1 - A note on graph pebbling

AU - Czygrinow, Andrzej

AU - Hurlbert, Glenn

AU - Kierstead, Henry

AU - Trotter, William T.

PY - 2002

Y1 - 2002

N2 - 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(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.

AB - 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(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.

KW - Connectivity

KW - Pebbling

KW - Threshold

UR - http://www.scopus.com/inward/record.url?scp=0036910031&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036910031&partnerID=8YFLogxK

U2 - 10.1007/s003730200015

DO - 10.1007/s003730200015

M3 - Article

VL - 18

SP - 219

EP - 225

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -