## 抄録

It has been shown that every quadrangulation on any non-spherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_{k} has chromatic number at least 4 if G has a cycle of odd length which cuts open N_{k} into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_{k} admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity.

本文言語 | English |
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ページ（範囲） | 100-114 |

ページ数 | 15 |

ジャーナル | Journal of Graph Theory |

巻 | 37 |

号 | 2 |

DOI | |

出版ステータス | Published - 2001 6月 |

## ASJC Scopus subject areas

- 幾何学とトポロジー