### Abstract

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this paper we tighten the gap between the upper and lower bounds for the pebbling threshold for the sequence of paths in the multiset model. We also find the pebbling threshold for the sequence of paths in the binomial model. Finally, we show that the spectrum of pebbling thresholds for graph sequences in the multiset model spans the entire range from n^{1 / 2} to n, answering a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remains unknown.

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

Pages (from-to) | 3297-3307 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 15 |

DOIs | |

State | Published - Aug 6 2008 |

### Fingerprint

### Keywords

- Paths
- Pebbling
- Spectrum
- Threshold

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*308*(15), 3297-3307. https://doi.org/10.1016/j.disc.2007.06.045

**On the pebbling threshold of paths and the pebbling threshold spectrum.** / Czygrinow, Andrzej; Hurlbert, Glenn H.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 308, no. 15, pp. 3297-3307. https://doi.org/10.1016/j.disc.2007.06.045

}

TY - JOUR

T1 - On the pebbling threshold of paths and the pebbling threshold spectrum

AU - Czygrinow, Andrzej

AU - Hurlbert, Glenn H.

PY - 2008/8/6

Y1 - 2008/8/6

N2 - A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this paper we tighten the gap between the upper and lower bounds for the pebbling threshold for the sequence of paths in the multiset model. We also find the pebbling threshold for the sequence of paths in the binomial model. Finally, we show that the spectrum of pebbling thresholds for graph sequences in the multiset model spans the entire range from n1 / 2 to n, answering a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remains unknown.

AB - A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this paper we tighten the gap between the upper and lower bounds for the pebbling threshold for the sequence of paths in the multiset model. We also find the pebbling threshold for the sequence of paths in the binomial model. Finally, we show that the spectrum of pebbling thresholds for graph sequences in the multiset model spans the entire range from n1 / 2 to n, answering a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remains unknown.

KW - Paths

KW - Pebbling

KW - Spectrum

KW - Threshold

UR - http://www.scopus.com/inward/record.url?scp=43249094978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43249094978&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2007.06.045

DO - 10.1016/j.disc.2007.06.045

M3 - Article

AN - SCOPUS:43249094978

VL - 308

SP - 3297

EP - 3307

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 15

ER -