Medial symmetry type graphs

TL;DR

This paper classifies symmetry types of k-orbit maps, especially for k up to 7, using symmetry type graphs, and explores their self-duality properties to extend previous classifications.

Contribution

It extends the classification of k-orbit maps to k=5 and beyond, including self-dual types, using symmetry type graphs, which was not previously done.

Findings

Classified all 5-orbit map types.

Extended classification to k ≤ 7.

Identified self-dual symmetry types.

Abstract

A $k$-orbit map is a map with its automorphism group partitioning the set of flags into $k$ orbits. Recently $k$-orbit maps were studied by Orbani\' c, Pellicer and Weiss, for $k \leq 4$. In this paper we use symmetry type graphs to extend such study and classify all the types of $5$-orbit maps, as well as all self-dual, properly and improperly, symmetry type of $k$-orbit maps with $k\leq 7$. Moreover, we determine, for small values of $k$, all types of $k$-orbits maps that are medial maps. Self-dualities constitute an important tool in this quest.