Planar cover

Additionally they show that the ply of a planar cover may be arbitrarily large.

Negami (1986) proved, conversely, that if a connected graph H has a two-ply planar cover then H must have an embedding into the projective plane.

Negami (1988) proved that every connected graph with a planar regular cover can be embedded into the projective plane.

[12] Based on these two results, he conjectured that in fact every connected graph with a planar cover is projective.

[14] It is also known as Negami's "1-2-∞ conjecture", because it can be reformulated as stating that the minimum ply of a planar cover, if it exists, must be either 1 or 2.

[16] Since Negami made his conjecture, it has been proven that 31 of these 32 forbidden minors either do not have planar covers, or can be reduced by Y-Δ transforms to a simpler graph on this list.

[21] The conjecture is true for regular emulators, coming from automorphisms of the underlying covering graph, by a result analogous to the result of Negami (1988) for regular planar covers.

[22] However, several of the 32 connected forbidden minors for projective-planar graphs turn out to have planar emulators.

Finding a full set of forbidden minors for the existence of planar emulators remains an open problem.

The graph C is a planar cover of the graph H . The covering map is indicated by the vertex colors.
Identifying pairs of opposite vertices of the dodecahedron gives a covering map to the Petersen graph
The dodecagonal prism can form a 2-ply cover of the hexagonal prism , a 3-ply cover of the cube , or a 4-ply cover of the triangular prism .
K 1,2,2,2 , the only possible minimal counterexample to Negami's conjecture