On Negami's planar cover conjecture
Yo'av Rieck, Yasushi Yamashita
Abstract
Given a finite cover f:tildeG G and an embedding of tildeG in the plane, Negami conjectures that G embeds in P2. Negami proved this conjecture for regular covers. In this paper we define two properties (Propserties V and E), depending on the cover tildeG and its embedding into S2, and generalize Negami's result by showing: (1) If Properties V and E are fulfilled then G embeds in P2. (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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