### 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) |
---|---|

Pages (from-to) | 93-105 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 247 |

Issue number | 1-3 |

State | Published - Mar 28 2002 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*247*(1-3), 93-105.

**On pebbling threshold functions for graph sequences.** / Czygrinow, Andrzej; Eaton, Nancy; Huribert, Glenn; Kayll, P. Mark.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 247, no. 1-3, pp. 93-105.

}

TY - JOUR

T1 - On pebbling threshold functions for graph sequences

AU - Czygrinow, Andrzej

AU - Eaton, Nancy

AU - Huribert, Glenn

AU - Kayll, P. Mark

PY - 2002/3/28

Y1 - 2002/3/28

N2 - 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>t0, 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.

AB - 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>t0, 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.

KW - Pebbling number

KW - S0012-365X(01)00163-7

KW - Threshold function

UR - http://www.scopus.com/inward/record.url?scp=31244437193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=31244437193&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:31244437193

VL - 247

SP - 93

EP - 105

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -