Pebbling in dense graphs

Andrzej Czygrinow, Glenn Hurlbert

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number π(G) so that every configuration of π(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n ≥ 9 vertices having minimum degree at least ⌊n/2⌋ are Class 0, as are bipartite graphs with m ≥ 336 vertices in each part having minimum degree at least ⌊m/2⌋ + 1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree δ, with √n ≪ δ, is O(n3/2/δ), which is tight when δ is proportional to n.

Original languageEnglish (US)
Pages (from-to)201-208
Number of pages8
JournalAustralasian Journal of Combinatorics
Volume28
StatePublished - Dec 1 2003

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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