20 Years of Negami's Planar Cover Conjecture

On Possible Counterexamples to Negami's Planar Cover Conjecture

On possible counterexamples to Negami's planar cover conjecture

A simple graph H is a cover of a graph G if there exists a mapping φ from H onto G such that φ maps the neighbors of every vertex v in H bijectively to the neighbors of φ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, a...

A note on possible extensions of Negami's conjecture

A graph H is a cover of a graph G if there exists a mapping φ from V (H) onto V (G) such that for every vertex v of G, φ maps the neighbours of v in H bijectively onto the neighbours of φ(v) in G. Negami conjectured in 1987 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. This conjecture is not completely solved yet, but partial results due to A...

Finite planar emulators for K4, 5-4K2 and K1, 2, 2, 2 and Fellows' Conjecture

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 K4,5 − 4K2. Archdeacon [Dan Archdeacon, Two graphswithout planar covers, J. Graph Theory, 41 (4) (2002) 318–326] showed that K4,5−4K2 does not admit a finite planar cover; thusK4,5−4K2 provides a counterexample to Fellows’...

m at h . C O ] 1 9 D ec 2 00 8 FINITE PLANAR EMULATORS FOR K 4 , 5 − 4 K

In 1988 M. 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 K4,5 − 4K2. D. Archdeacon [2] showed that K4,5 − 4K2 does not admit a finite planar cover; thus K4,5 − 4K2 provides a counterexample to Fellows’ Conjecture. It is known that S. Negami’s Planar Cover Conjecture is true i...