# Optimal pebbling number of graphs with given minimum degree

Andrzej Czygrinow, G. Hurlbert, G. Y. Katona, L. F. Papp

Research output: Contribution to journalArticle

### 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) Discrete Applied Mathematics https://doi.org/10.1016/j.dam.2019.01.023 Published - Jan 1 2019

### Fingerprint

Minimum Degree
Graph in graph theory
Vertex of a graph
Connected graph
Equality
Arbitrary

### Keywords

• Given minimum degree
• Graph pebbling
• Optimal pebbling

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Applied Mathematics

### Cite this

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

In: Discrete Applied Mathematics, 01.01.2019.

Research output: Contribution to journalArticle

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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].

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