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 languageEnglish (US)
JournalDiscrete Applied Mathematics
DOIs
StatePublished - Jan 1 2019

Fingerprint

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

@article{736b9d2d730349498da98807cc52b268,
title = "Optimal pebbling number of graphs with given minimum degree",
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].",
keywords = "Given minimum degree, Graph pebbling, Optimal pebbling",
author = "Andrzej Czygrinow and G. Hurlbert and Katona, {G. Y.} and Papp, {L. F.}",
year = "2019",
month = "1",
day = "1",
doi = "10.1016/j.dam.2019.01.023",
language = "English (US)",
journal = "Discrete Applied Mathematics",
issn = "0166-218X",
publisher = "Elsevier",

}

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 -