### Abstract

Given a connected graph G, and a distribution of / pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number /, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs 'S = (Gi,G2,...,G,...), where G has n vertices, is any function ta(n) such that almost all distributions of / pebbles are solvable when t>t_{0}, and such that almost none are solvable when t<$to. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.

Original language | English (US) |
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Pages (from-to) | 93-105 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 247 |

Issue number | 1-3 |

DOIs | |

State | Published - Mar 28 2002 |

### Keywords

- Pebbling number
- S0012-365X(01)00163-7
- Threshold function

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*247*(1-3), 93-105. https://doi.org/10.1016/S0012-365X(01)00163-7