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arXiv · 2412.19560

Three Theorems on Negami's Planar Cover Conjecture

Dickson Annor·Yuri Nikolayevsky·Michael Payne
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Abstract

A long-standing Conjecture of S. Negami states that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It is known that the Conjecture is equivalent to the fact that \emph{the graph $K_{1,2, 2, 2}$ has no finite planar cover}. We prove three theorems showing that the graph $K_{1,2, 2, 2}$ admits no planar cover with certain structural properties, and that the minimal planar cover of $K_{1,2, 2, 2}$ (if it exists) must be $4$-connected.

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Dickson Annor, Yuri Nikolayevsky, Michael Payne. 2026-07-17. Three Theorems on Negami's Planar Cover Conjecture. https://arxiv.org/abs/2412.19560

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