Another two graphs with no planar covers (Q2746205)

In 1986 Negami conjectured that a connected graph has a finite planar cover if and only if it imbeds in the projective plane. (Familiar examples include the icosahedron double covering the complete graph \(K_6\) in the projective plane and, dually, the dodecahedron double covering the Petersen graph.) The graph \(K_{1,2,2,2}\) does not embed in the projective plane. If it could be shown that this graph does not have a finite planar cover, then Negami's conjecture would follow. But that question is open. In the present paper the author shows that two nonprojective graphs related to \(K_{1,2,2,2}\) (by replacing either two vertex-disjoint triangles, or four edge-disjoint triangles, not incident with the vertex of degree 6, with cubic vertices) have no planar covers. This result is used in a subsequent paper to limit the number of possible counterexamples to the conjecture of Negami.