## Abstract

Consider a distribution of pebbles on a connected graph G. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number π ^{∗} (G) is the smallest number of pebbles that we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most [Formula presented], where δ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If diam(G)≥3 then we further improve the bound to π ^{∗} (G)≤[Formula presented]. On the other hand, we show that, for arbitrary large diameter and any ϵ>0, there are infinitely many graphs whose optimal pebbling number is bigger than [Formula presented]−ϵ[Formula presented].

Original language | English (US) |
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Pages (from-to) | 117-130 |

Number of pages | 14 |

Journal | Discrete Applied Mathematics |

Volume | 260 |

DOIs | |

State | Published - May 15 2019 |

## Keywords

- Given minimum degree
- Graph pebbling
- Optimal pebbling

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics