### Abstract

The pebbling number of a graph is the smallest number t such that from any initial configuration of t pebbles one can move a pebble to any prescribed vertex by a sequence of pebbling steps. It is known that graphs whose connectivity is high compared to their diameter have a pebbling number as small as possible. We will use the above result to prove two related theorems. First, answering a question of the second author, we show that there exist graphs of arbitrarily high constant girth and least possible pebbling number. In the second application, we prove that the product of two graphs of high minimum degree has a pebbling number equal to the number of vertices of the product. This shows that Graham's product conjecture is true in the case of high minimum degree graphs. In addition, we consider a probabilistic variant of the pebbling problem and establish a pebbling threshold result for products of paths. The last result shows that the sequence of paths satisfies the probabilistic analogue of Graham's product conjecture.

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

Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - 2006 |

### Fingerprint

### Keywords

- Connectivity
- Girth
- Grids
- Pebbling
- Threshold

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*SIAM Journal on Discrete Mathematics*,

*20*(1), 1-10. https://doi.org/10.1137/S0895480102416374

**Girth, pebbling, and grid thresholds.** / Czygrinow, Andrzej; Hurlbert, Glenn.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 20, no. 1, pp. 1-10. https://doi.org/10.1137/S0895480102416374

}

TY - JOUR

T1 - Girth, pebbling, and grid thresholds

AU - Czygrinow, Andrzej

AU - Hurlbert, Glenn

PY - 2006

Y1 - 2006

N2 - The pebbling number of a graph is the smallest number t such that from any initial configuration of t pebbles one can move a pebble to any prescribed vertex by a sequence of pebbling steps. It is known that graphs whose connectivity is high compared to their diameter have a pebbling number as small as possible. We will use the above result to prove two related theorems. First, answering a question of the second author, we show that there exist graphs of arbitrarily high constant girth and least possible pebbling number. In the second application, we prove that the product of two graphs of high minimum degree has a pebbling number equal to the number of vertices of the product. This shows that Graham's product conjecture is true in the case of high minimum degree graphs. In addition, we consider a probabilistic variant of the pebbling problem and establish a pebbling threshold result for products of paths. The last result shows that the sequence of paths satisfies the probabilistic analogue of Graham's product conjecture.

AB - The pebbling number of a graph is the smallest number t such that from any initial configuration of t pebbles one can move a pebble to any prescribed vertex by a sequence of pebbling steps. It is known that graphs whose connectivity is high compared to their diameter have a pebbling number as small as possible. We will use the above result to prove two related theorems. First, answering a question of the second author, we show that there exist graphs of arbitrarily high constant girth and least possible pebbling number. In the second application, we prove that the product of two graphs of high minimum degree has a pebbling number equal to the number of vertices of the product. This shows that Graham's product conjecture is true in the case of high minimum degree graphs. In addition, we consider a probabilistic variant of the pebbling problem and establish a pebbling threshold result for products of paths. The last result shows that the sequence of paths satisfies the probabilistic analogue of Graham's product conjecture.

KW - Connectivity

KW - Girth

KW - Grids

KW - Pebbling

KW - Threshold

UR - http://www.scopus.com/inward/record.url?scp=33847662088&partnerID=8YFLogxK

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U2 - 10.1137/S0895480102416374

DO - 10.1137/S0895480102416374

M3 - Article

AN - SCOPUS:33847662088

VL - 20

SP - 1

EP - 10

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -