Three Theorems on Negami's Planar Cover Conjecture

TL;DR

This paper investigates Negami's Planar Cover Conjecture by proving that the specific graph K_{1,2,2,2} cannot have certain types of finite planar covers and that any minimal cover must be 4-connected.

Contribution

It provides three theorems that restrict the structural possibilities of planar covers for K_{1,2,2,2} and shows the minimal cover, if it exists, must be 4-connected.

Findings

K_{1,2,2,2} admits no planar cover with certain properties

Any minimal planar cover of K_{1,2,2,2} must be 4-connected

Supports the conjecture by narrowing the conditions for potential counterexamples

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.