### Abstract

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham's conjecture does not hold for rubbling numbers.

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

Pages (from-to) | 3436-3446 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 10 |

DOIs | |

State | Published - May 28 2009 |

### Fingerprint

### Keywords

- Optimal pebbling
- Pebbling
- Rubbling

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*309*(10), 3436-3446. https://doi.org/10.1016/j.disc.2008.09.035

**Rubbling and optimal rubbling of graphs.** / Belford, Christopher; Sieben, Nandor.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 309, no. 10, pp. 3436-3446. https://doi.org/10.1016/j.disc.2008.09.035

}

TY - JOUR

T1 - Rubbling and optimal rubbling of graphs

AU - Belford, Christopher

AU - Sieben, Nandor

PY - 2009/5/28

Y1 - 2009/5/28

N2 - A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham's conjecture does not hold for rubbling numbers.

AB - A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham's conjecture does not hold for rubbling numbers.

KW - Optimal pebbling

KW - Pebbling

KW - Rubbling

UR - http://www.scopus.com/inward/record.url?scp=67349112995&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349112995&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2008.09.035

DO - 10.1016/j.disc.2008.09.035

M3 - Article

VL - 309

SP - 3436

EP - 3446

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 10

ER -