Girth, pebbling, and grid thresholds

Andrzej Czygrinow, Glenn Hurlbert

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

The pebbling number of a graph is the smallest number t such that from any initial configuration of t pebbles one can move a pebble to any prescribed vertex by a sequence of pebbling steps. It is known that graphs whose connectivity is high compared to their diameter have a pebbling number as small as possible. We will use the above result to prove two related theorems. First, answering a question of the second author, we show that there exist graphs of arbitrarily high constant girth and least possible pebbling number. In the second application, we prove that the product of two graphs of high minimum degree has a pebbling number equal to the number of vertices of the product. This shows that Graham's product conjecture is true in the case of high minimum degree graphs. In addition, we consider a probabilistic variant of the pebbling problem and establish a pebbling threshold result for products of paths. The last result shows that the sequence of paths satisfies the probabilistic analogue of Graham's product conjecture.

Original languageEnglish (US)
Pages (from-to)1-10
Number of pages10
JournalSIAM Journal on Discrete Mathematics
Volume20
Issue number1
DOIs
StatePublished - Dec 1 2006

Keywords

  • Connectivity
  • Girth
  • Grids
  • Pebbling
  • Threshold

ASJC Scopus subject areas

  • Mathematics(all)

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