### 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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Journal | Discrete Applied Mathematics |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Given minimum degree
- Graph pebbling
- Optimal pebbling

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2019.01.023

**Optimal pebbling number of graphs with given minimum degree.** / Czygrinow, Andrzej; Hurlbert, G.; Katona, G. Y.; Papp, L. F.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2019.01.023

}

TY - JOUR

T1 - Optimal pebbling number of graphs with given minimum degree

AU - Czygrinow, Andrzej

AU - Hurlbert, G.

AU - Katona, G. Y.

AU - Papp, L. F.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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].

AB - 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].

KW - Given minimum degree

KW - Graph pebbling

KW - Optimal pebbling

UR - http://www.scopus.com/inward/record.url?scp=85061569208&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061569208&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2019.01.023

DO - 10.1016/j.dam.2019.01.023

M3 - Article

AN - SCOPUS:85061569208

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -