Introducing edge-biregular maps

2020/10/29 by Olivia Jeans, Jeans, Olivia Reade
Mathematics · #05C25 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2010.15743

openalex publication_date 2020/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the concept of alternate-edge-colourings for maps, and study highly symmetric examples of such maps. Edge-biregular maps of type (k,l) occur as smooth normal quotients of a particular index two subgroup of T_k,l, the full triangle group describing regular plane (k,l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.

Citations

• Automorphism groups of edge-transitive maps

Related

• Classifying edge-biregular maps of negative prime Euler characteristic
• A classification of vertex-reversing maps with Euler characteristic coprime to the edge number
• Parallel-product decomposition of edge-transitive maps
• Bicolored tilings and the Scott map
• Dihedral and cyclic symmetric maps on surfaces