## Abstract

It is proved that there is a function f: ℕ → ℕ such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ≥ f(g). Then (i) If every contractible 4-cycle of G is facial and there is a face of size > 4, then G is 3-colorable. (ii) If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C_{1}, ⋯, C_{g} such that, after cutting along C_{1}, ⋯, C_{g}, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite. If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature.

Original language | English (US) |
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Pages (from-to) | 301-310 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 84 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2002 |

## All Science Journal Classification (ASJC) codes

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