## Abstract

The Tower of Hanoi problem is generalized in such a way that the pegs are located at the vertices of a directed graph G, and moves of disks may be made only along edges of G. Leiss obtained a complete characterization of graphs in which arbitrarily many disks can be moved from the source vertex S to the destination vertex D. Here we consider graphs which do not satisfy this characterization; hence, there is a bound on the number of disks which can be handled. Denote by g_{n} the maximal such number as G varies over all such graphs with n vertices and S, D vary over the vertices. Answering a question of Leiss [Finite Hanoi problems: How many discs can be handled? Congr. Numer. 44 (1984) 221-229], we prove that g_{n} grows sub-exponentially fast. Moreover, there exists a constant C such that g_{n} ≤ Cn^{1 / 2 log2 n} for each n. On the other hand, for each ε > 0 there exists a constant C_{ε} > 0 such that g_{n} ≥ C_{ε} n^{(1 / 2 - ε) log2 n} for each n.

Original language | English |
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Pages (from-to) | 377-383 |

Number of pages | 7 |

Journal | Theoretical Computer Science |

Volume | 369 |

Issue number | 1-3 |

DOIs | |

State | Published - 15 Dec 2006 |

## Keywords

- Digraph
- Graphs
- Hanoi Tower problem
- Recurrence equation

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science