# Regular maps of order $2$-powers

## TL;DR

This paper investigates the existence and classification of regular maps of order 2^n, focusing on types defined by valency and covalency, and extends known classifications for larger n with new conjectures and confirmations.

## Contribution

It extends the classification of regular maps of order 2^n for n ≥ 12, introduces new existence results for certain types, and proposes a conjecture about non-existence in specific cases.

## Key findings

- Existence of regular maps with types where s+t ≤ n or s=t.
- Classification of regular maps with types {2^{n-2}, 2^{n-2}} and {2^{n-3}, 2^{n-3}}.
- Conjecture that no regular maps exist for s+t > n with s<t, confirmed for t=n-2 and n-3.

## Abstract

In this paper, we consider the possible types of regular maps of order $2^n$, where the order of a regular map is the order of automorphism group of the map. For $n \le 11$, M. Conder classified all regular maps of order $2^n$. It is easy to classify regular maps of order $2^n$ whose valency or covalency is $2$ or $2^{n-1}$. So we assume that $n \geq 12$ and $2\leq s,t\leq n-2$ with $s\leq t$ to consider regular maps of order $2^n$ with type $\{2^s, 2^t\}$. We show that for $s+t\leq n$ or for $s+t>n$ with $s=t$, there exists a regular map of order $2^n$ with type $\{2^s, 2^t\}$, and furthermore, we classify regular maps of order $2^n$ with types $\{2^{n-2},2^{n-2}\}$ and $\{2^{n-3},2^{n-3}\}$. We conjecture that, if $s+t>n$ with $s<t$, then there is no regular map of order $2^n$ with type $\{2^s, 2^t\}$, and we confirm the conjecture for $t=n-2$ and $n-3$.