A classification of vertex-reversing maps with Euler characteristic coprime to the edge number

Abstract

An arc-regular map is \emph{vertex-reversing} if the automorphism group has dihedral vertex stabilizers. This paper classifies solvable vertex-reversing maps whose Euler characteristic is coprime to the edge number. The classification establishes that such maps fall into four families: $\D_{2n}$-maps, $(\D_{2m}\times\D_{2n})$-maps, $(\ZZ_{mn\ell}{:}\D_4)$-maps, and $(\ZZ_{3^f n}.\S_4)$-maps, with the parameters specified in the main theorem. Moreover, for each family, we provide an explicit formula for the Euler characteristic.