Suppose that on a surface with Euler characteristicamap withfacescan be coloured by at mostcolourswhere

Write this inequality as

Ifthensome faces can be coloured twice only if they have no edges incommon. We consider the least value offorvalues of

Ifthensothat at least 7 colours are needed.

If %XHI =1 then

Proceeding in this way, we obtain the table

2 | 6 |

1 | 6 |

0 | 7 |

-1 | 7 |

-2 | 8 |

-3 | 9 |

-4 | 9 |

-5 | 10 |

-6 | 10 |

-7 | 10 |

-8 | 11 |

-9 | 11 |

-10 | 12 |