### 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 each 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 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 give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.

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

Pages (from-to) | 535-551 |

Number of pages | 17 |

Journal | Graphs and Combinatorics |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - May 2013 |

### Fingerprint

### Keywords

- Pebbling
- Rubbling

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

*29*(3), 535-551. https://doi.org/10.1007/s00373-012-1146-2

**Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs.** / Katona, Gyula Y.; Sieben, Nandor.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 29, no. 3, pp. 535-551. https://doi.org/10.1007/s00373-012-1146-2

}

TY - JOUR

T1 - Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs

AU - Katona, Gyula Y.

AU - Sieben, Nandor

PY - 2013/5

Y1 - 2013/5

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 each 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 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 give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.

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 each 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 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 give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.

KW - Pebbling

KW - Rubbling

UR - http://www.scopus.com/inward/record.url?scp=84877598748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877598748&partnerID=8YFLogxK

U2 - 10.1007/s00373-012-1146-2

DO - 10.1007/s00373-012-1146-2

M3 - Article

VL - 29

SP - 535

EP - 551

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 3

ER -