An improved upper bound for the pebbling threshold of the n-path

Given a configuration of t indistinguishable pebbles on the n vertices of a graph G, we say that a vertex v can be reached if a pebble can be placed on it in a finite number of “moves”. G is said to be pebbleable if all its vertices can be thus reached. Now given the n-path Pn how large (resp. small...

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Published inDiscrete mathematics Vol. 275; no. 1-3; pp. 367 - 373
Main Authors Wierman, Adam, Salzman, Julia, Jablonski, Michael, Godbole, Anant P.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 28.01.2004
Elsevier
Subjects
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Mathematics
n-Cycle
n-Path
Pebbling number
Pebbling threshold
Sciences and techniques of general use
n-Path
Pebbling number
Pebbling threshold
n-Cycle
n cycle
Upper bound
Discrete mathematics
Graph theory
n path
Threshold function
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Summary:Given a configuration of t indistinguishable pebbles on the n vertices of a graph G, we say that a vertex v can be reached if a pebble can be placed on it in a finite number of “moves”. G is said to be pebbleable if all its vertices can be thus reached. Now given the n-path Pn how large (resp. small) must t be so as to be able to pebble the path almost surely (resp. almost never)? It was known that the threshold th(Pn) for pebbling the path satisfies n2clgn⩽th(Pn)⩽n22lgn, where lg=log2 and c<1/2 is arbitrary. We improve the upper bound for the threshold function to th(Pn)⩽n2dlgn, where d>1 is arbitrary.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2002.10.001