### Abstract

Let G be a 3-connected bipartite graph with partite sets X ∪ Y which is embeddable in the torus. We shall prove that G has a Hamiltonian cycle if (i) G is balanced, i.e., |X| = |Y|, and (ii) each vertex x∈ X has degree four. In order to prove the result, we establish a result on orientations of quadrangular torus maps possibly with multiple edges. This result implies that every 4-connected toroidal graph with toughness exactly one is Hamiltonian, and partially solves a well-known Nash-Williams' conjecture.

Original language | English |
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Pages (from-to) | 46-60 |

Number of pages | 15 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 103 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

### Keywords

- Bipartite graph
- Hamiltonian cycle
- Quadrangulation
- Torus

### ASJC Scopus subject areas

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

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## Cite this

Fujisawa, J., Nakamoto, A., & Ozeki, K. (2013). Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices.

*Journal of Combinatorial Theory. Series B*,*103*(1), 46-60. https://doi.org/10.1016/j.jctb.2012.08.004