Graph Theory

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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 phi from H onto G such that phi maps the neighbors of every vertex v in H bijectively to the neighbors of phi(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, and the first author that the conjecture holds as long as the graph K-1,K-2,K-2,K-2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list