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A classification of rotary embeddings of multicycles

Zhaochen Ding, Zheng Guo, Luyi Liu

Abstract

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle $\mathrm{C}_n^{(λ)}$ of length $n$ and edge-multiplicity $λ$, we determine all rotary embeddings for $n\geqslant 3$ and $λ\geqslant 2$. When $n$ is odd, there is a unique isomorphism class; when $n$ is even, the embeddings form a family $\mathcal{M}_n^{(λ)}(i,j)$ parameterized by integer pairs $(i,j)$ satisfying explicit congruence conditions.

A classification of rotary embeddings of multicycles

Abstract

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle of length and edge-multiplicity , we determine all rotary embeddings for and . When is odd, there is a unique isomorphism class; when is even, the embeddings form a family parameterized by integer pairs satisfying explicit congruence conditions.
Paper Structure (6 sections, 8 theorems, 10 equations)

This paper contains 6 sections, 8 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Rotary maps as coset configurations
  3. Arc-transitive embeddings of multicycles
  4. Proof of Theorem \ref{['thm:main']}
  5. Odd lengths
  6. Even lengths

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a rotary embedding of $\mathrm{C}_n^{(\lambda)}$ with $n\geqslant 3$ and $\lambda\geqslant 2$. Then either

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.4
  • ...and 5 more