$K_{1,2,2,2}$ has no $n$-fold planar cover graph for $n<14$

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. Hlin\v{e}n\'{y}, 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 $14$.