Arc-transitive maps with edge number coprime to the Euler characteristic -- I
Cai Heng Li, Luyi Liu
TL;DR
The paper advances a classification of arc-transitive maps with $\gcd(|E|, \chi(\mathcal{M}))=1$ under solvable automorphism groups by organizing the problem into Almost Sylow-cyclic (ASG) categories and analyzing metacyclic, 2-nilpotent, and other structured groups. It develops a systematic framework using coset-graph representations and rotary/reversing embeddings to derive explicit forms, counts of inequivalent data, and concrete graph types such as $\mathrm{K}_2^{(m)}$, $\mathrm{C}_m$, and $\mathrm{K}_4^{(\lambda)}$. The main contributions include explicit solvable-case classifications (Types I–V), detailed descriptions of rotary pairs and reversing triples, and constructive examples (notably Type IV and Type V) that illustrate the rich combinatorial and group-theoretic structure of arc-transitive maps with coprime edge numbers and Euler characteristics. This work lays a solid foundation for the broader program of classifying all arc-transitive maps with $\gcd(|E|, \chi(\mathcal{M}))=1$, with potential implications for graph embeddings, symmetry analysis, and the enumeration of highly symmetric maps on surfaces.
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.