### Abstract

The square of a path (cycle) is the graph obtained by joining every pair of vertices of distance two in the path (cycle). Let G be a graph on n vertices with minimum degree δ(G). Posá conjectured that if δ(G) ≥ 2 3n, then G contains the square of a hamiltonian cycle. This is also a special case-of a conjecture of Seymour. In this paper, we prove that for any ε > 0, there exists a number m, depending only on ε, such that if δ(G) ≥ (2/3 + ε) n + m, then G contains the square of a hamitonian path between any two edges, which implies the squares of a hamiltonian cycle.

Original language | English (US) |
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Pages (from-to) | 55-64 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 63 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1995 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Fan, G. H., & Kierstead, H. (1995). The square of paths and cycles.

*Journal of Combinatorial Theory, Series B*,*63*(1), 55-64. https://doi.org/10.1006/jctb.1995.1005