Title: |
Removing Even Crossings, Continued |

Authors: |
Michael J. Pelsmajer, Marcus Schaefer, Daniel Štefankovič |

Abstract: |
In this paper we investigate how certain results related
to the Hanani-Tutte theorem can be lifted to orientable surfaces of
higher genus. We give a new simple, geometric proof that the weak
Hanani-Tutte theorem is true for higher-genus surfaces. We extend the
proof to prove that bipartite generalized thrackles in a surface S
can be embedded in S.We also show that a result of Pach and Tòth that allows the redrawing of a graph removing intersections on even edges remains true on higher-genus surfaces. As a consequence, we can conclude that cr _{S}(G), the crossing number
of the graph G on surface S, is bounded by 2 ocr_{S}(G),
where ocr_{S}(G) is the odd crossing number
of G on surface S.Finally, we begin an investigation of optimal crossing configurations for which cr is linearly bounded in ocr. |

Full Paper: |
[postscript, pdf] |