On pebbling threshold functions for graph sequences

Andrzej Czygrinow, Nancy Eaton, Glenn Huribert, P. Mark Kayll

Research output: Contribution to journalArticle

13 Scopus citations

Abstract

Given a connected graph G, and a distribution of / pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number /, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs 'S = (Gi,G2,...,G,...), where G has n vertices, is any function ta(n) such that almost all distributions of / pebbles are solvable when t>t0, and such that almost none are solvable when t<$to. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.

Original languageEnglish (US)
Pages (from-to)93-105
Number of pages13
JournalDiscrete Mathematics
Volume247
Issue number1-3
DOIs
StatePublished - Mar 28 2002

Keywords

  • Pebbling number
  • S0012-365X(01)00163-7
  • Threshold function

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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