A classification of rotary embeddings of multicycles

TL;DR

This paper classifies all rotary embeddings of multicycles, providing a complete description of their structure depending on cycle length and edge multiplicity, with explicit parameterizations for even cycles.

Contribution

It offers a comprehensive classification of rotary embeddings of multicycles, including explicit conditions and parameterizations, extending previous understanding of these graph embeddings.

Findings

Unique embedding for odd cycles

Parameterized family of embeddings for even cycles

Explicit congruence conditions for embeddings

Abstract

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