Archdeacon, D.; Bonnington, P. and Siran, J.
(2001).
*Journal of Graph Theory*, 38(4) pp. 230–243.

DOI (Digital Object Identifier) Link: | http://dx.doi.org/10.1002/jgt.10009 |
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## Abstract

Let ck = crk (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence co, c1,c2encodes the trade-off between adding handles and decreasing crossings. We focus on sequences of the type co > c1 > c2 = 0; equivalently, we study the planar and toroidal crossing number of doubly-toroidal graphs. For every > 0 we construct graphs whose orientable crossing sequence satisfies c1/co > 5/6-. In other words, we construct graphs where the addition of one handle can save roughly 1/6th of the crossings, but the addition of a second handle can save five times more crossings. We similarly define the non-orientable crossing sequence 0,1,2, ··· for drawings on non-orientable surfaces. We show that for every 0 > 1 > 0 there exists a graph with non-orientable crossing sequence 0, 1, 0. We conjecture that every strictly-decreasing sequence of non-negative integers can be both an orientable crossing sequence and a non-orientable crossing sequence (with different graphs).

Item Type: | Journal Article |
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ISSN: | 0364-9024 |

Keywords: | graph embeddings; crossing number |

Academic Unit/Department: | Mathematics, Computing and Technology > Mathematics and Statistics Mathematics, Computing and Technology |

Item ID: | 8099 |

Depositing User: | Jozef Širáň |

Date Deposited: | 14 Jun 2007 |

Last Modified: | 14 Jan 2016 16:32 |

URI: | http://oro.open.ac.uk/id/eprint/8099 |

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