### Abstract

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.

Original language | English |
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Pages (from-to) | 100-114 |

Number of pages | 15 |

Journal | Journal of Graph Theory |

Volume | 37 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2001 Jun |

### Keywords

- Chromatic number
- Eulerian quadrangulations
- Representativity

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Journal of Graph Theory*,

*37*(2), 100-114. https://doi.org/10.1002/jgt.1005