## Abstract

Consider the graph with all permutations of a symbol sequence as vertices, where two permutations are connected by an edge when they differ by an interchange of two distinct adjacent symbols. In 1965, D. H. Lehmer conjectured that all vertices in this graph can be visited by a Hamiltonian path that is possibly imperfect, in the sense of having spurs. Such a spur visits a vertex twice, with a single vertex in-between. We prove Lehmer’s conjecture for binary permutations that involve only two distinct symbols. For general symbol sequences, we identify the stutter permutations as candidate spur tips, and prove that the non-stutter permutations admit a disjoint cycle cover. We also provide new (simpler) proofs for some known results.

Original language | English |
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Pages (from-to) | 295-310 |

Number of pages | 16 |

Journal | Designs, Codes and Cryptography |

Volume | 84 |

Issue number | 1-2 |

Early online date | 14 Nov 2016 |

DOIs | |

Publication status | Published - 1 Jul 2017 |

## Keywords

- Hamiltonian path
- Minimal-change generation
- Permutation