K_1,2,2,2 has no n-fold planar cover graph for n<14

2023/11/03 by Annor, Dickson Y. B., Nikolayevsky, Yuri, Payne, Michael S.
#05C10 (Primary) #57M15 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.01672

Abstract

S. Negami conjectured in 1988 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It follows from the works of D. Archdeacon, M. Fellows, P. Hliněný, and S. Negami that this conjecture is true if the graph K_{1, 2, 2, 2} has no finite planar cover. We prove a number of structural results about putative finite planar covers of K_1,2,2,2 that may be of independent interest. We then apply these results to prove that K_{1, 2, 2, 2} has no planar cover of fold number less than 14.

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