## 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(n^{3/2}/δ), which is tight when δ is proportional to n.

Original language | English (US) |
---|---|

Pages (from-to) | 201-208 |

Number of pages | 8 |

Journal | Australasian Journal of Combinatorics |

Volume | 28 |

State | Published - Dec 1 2003 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics