### 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.

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

Pages (from-to) | 487-492 |

Number of pages | 6 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 38 |

DOIs | |

State | Published - Dec 1 2011 |

### Fingerprint

### Keywords

- Bounded diameter
- Pebbling
- Rubbling

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

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

Research output: Contribution to journal › Article

*Electronic Notes in Discrete Mathematics*, vol. 38, pp. 487-492. https://doi.org/10.1016/j.endm.2011.09.079

}

TY - JOUR

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

AU - Katona, Gyula Y.

AU - Sieben, Nandor

PY - 2011/12/1

Y1 - 2011/12/1

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.

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.

KW - Bounded diameter

KW - Pebbling

KW - Rubbling

UR - http://www.scopus.com/inward/record.url?scp=82955192408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82955192408&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2011.09.079

DO - 10.1016/j.endm.2011.09.079

M3 - Article

AN - SCOPUS:82955192408

VL - 38

SP - 487

EP - 492

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -