Arc-transitive maps with edge number coprime to the Euler characteristic -- I

TL;DR

This paper develops a framework for classifying arc-transitive maps with solvable automorphism groups where the edge number and Euler characteristic are coprime, revealing how edge number influences map properties.

Contribution

It introduces a classification framework for arc-transitive maps with coprime edge number and Euler characteristic, focusing on maps with solvable automorphism groups.

Findings

Constructed new arc-regular maps with specific properties

Established relationships between edge number and Euler characteristic

Provided a classification scheme for a family of maps

Abstract

This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one establishes a framework and carries out the classification work for arc-transitive maps with solvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the constructions of various new and interesting arc-regular maps.