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On Negami's planar cover conjecture

Yo'av Rieck·Yasushi Yamashita
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Abstract

Given a finite cover f:tilde{G} \to G and an embedding of tilde{G} in the plane, Negami conjectures that G embeds in P^2. Negami proved this conjecture for regular covers. In this paper we define two properties (Propserties V and E), depending on the cover tilde{G} and its embedding into S^2, and generalize Negami's result by showing: (1) If Properties V and E are fulfilled then G embeds in P^2. (2) Regular covers always fulfill Properties V and E. We give an example of an irregular cover fulfilling Properties V and E. Covers not fulfilling Properties V and E are discussed as well.

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Yo'av Rieck, Yasushi Yamashita. 2006-12-13. On Negami's planar cover conjecture. https://arxiv.org/abs/math/0612342

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