Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture

TL;DR

This paper constructs finite planar emulators for specific graphs, providing counterexamples to Fellows' Conjecture and advancing understanding of planar covers and emulators in graph theory.

Contribution

It presents the first known finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2}, challenging existing conjectures and highlighting differences between emulators and covers.

Findings

K_{4,5} - 4K_2 admits a finite planar emulator but no finite planar cover.

A finite planar emulator for K_{1,2,2,2} is constructed, but its cover status remains open.

Counterexamples to Fellows' Conjecture are provided through these constructions.

Abstract

In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open.