Three Theorems on Negami's Planar Cover Conjecture

2024/12/27 by Dickson Annor, Dickson Y. B. Annor, Annor, Dickson +5
Engineering · Mathematics · Computer Science · #Advanced Materials and Mechanics #Geometric and Algebraic Topology #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.2412.19560

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 \emphthe 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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