Finite planar emulators for K4,5−4K2K4,5−4K2 and K1,2,2,2K1,2,2,2 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 K4,5−4K2K4,5−4K2. Archdeacon [Dan Archdeacon, Two graphs without planar covers, J. Graph Theory, 41 (4) (2002) 318–326] showed that K4,5−4K2K4,5−4K2 does not admit a finite planar cover; thus K4,5−4K2K4,5−4K2 provides a counterexample to Fellows’ Conjecture.It is known that Negami’s Planar Cover Conjecture is true if and only if K1,2,2,2K1,2,2,2 admits no finite planar cover. We construct a finite planar emulator for K1,2,2,2K1,2,2,2. The existence of a finite planar cover for K1,2,2,2K1,2,2,2 is still open.