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Cayley hyper-digraphs and Cayley hypermaps

Kai Yuan, Yan Wang

TL;DR

This work develops a general framework for Cayley hyper-digraphs and Cayley hypermaps, aiming to understand highly symmetric hypergraph-based embeddings. It defines Cayley hyper-digraphs $CD(G,X)$ and shows how a regular $G$-action yields vertex-transitive structures, with a normalizer $G_R\rtimes Aut(G,X)$ governing automorphisms. It then introduces ideal cycles and ideal Cayley hypergraphs to construct Cayley hypermaps $CM(G,Y,\rho_{[Y]},\tau_{[r]})$, including explicit examples like the Fano plane realized on the torus and on genus-3 surfaces. The paper provides concrete criteria for automorphism groups, including a formula $Aut(\mathcal{M})=Aut(\mathcal{M})_{1_G}G_R$ with a cyclic stabilizer, and outlines directions for classification and exploration of regular and non-orientable cases. Overall, the results establish a versatile theory linking Cayley structures to hypermap embeddings and symmetry analysis, enabling systematic construction and analysis of highly symmetric hypermaps.

Abstract

A Cayley hyper-digraph is a directed hypergraph that its automorphism group contains a subgroup acting regularly on vertices and a Cayley hypermap is a hypermap whose automorphism group contains a subgroup which induces regular action on the hypervertex set. In this paper, we study Cayley hyper-digraphs and construct Cayley hypermaps which have high level of symmetry. Our main goal is to present the general theory so as to make it clear to study Cayley hypermaps.

Cayley hyper-digraphs and Cayley hypermaps

TL;DR

This work develops a general framework for Cayley hyper-digraphs and Cayley hypermaps, aiming to understand highly symmetric hypergraph-based embeddings. It defines Cayley hyper-digraphs and shows how a regular -action yields vertex-transitive structures, with a normalizer governing automorphisms. It then introduces ideal cycles and ideal Cayley hypergraphs to construct Cayley hypermaps , including explicit examples like the Fano plane realized on the torus and on genus-3 surfaces. The paper provides concrete criteria for automorphism groups, including a formula with a cyclic stabilizer, and outlines directions for classification and exploration of regular and non-orientable cases. Overall, the results establish a versatile theory linking Cayley structures to hypermap embeddings and symmetry analysis, enabling systematic construction and analysis of highly symmetric hypermaps.

Abstract

A Cayley hyper-digraph is a directed hypergraph that its automorphism group contains a subgroup acting regularly on vertices and a Cayley hypermap is a hypermap whose automorphism group contains a subgroup which induces regular action on the hypervertex set. In this paper, we study Cayley hyper-digraphs and construct Cayley hypermaps which have high level of symmetry. Our main goal is to present the general theory so as to make it clear to study Cayley hypermaps.

Paper Structure

This paper contains 9 sections, 23 theorems, 57 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Cayley hyper-digraphs
  3. Basic Properties
  4. Automorphisms of Cayley hyper-digraphs
  5. Cayley hypermaps
  6. Ideal cycles
  7. Introducing Cayley hypermaps
  8. Automorphism group of Cayley hypermaps
  9. Further research

Key Result

Theorem 2.1

Let $\mathcal{H}=(V, D)$ be a hyper-digraph and $\varGamma=(V,E)$ the corresponding hypergraph, where $E=\{ \mathbf{x} \bigm| (u, \mathbf{x})\in D, \, {\rm for\, some}\, u\in V\}$. Suppose that a subgroup $G\le \hbox{\rm Aut,}(\mathcal{H})$ acts regularly on hypervertices. Pick up an $v\in V$ and wr

Figures (5)

  • Figure 1: $\hbox{\rm CH}(\mathbb{Z}_8,Y)$ drawn on Projective Plane with $Y=\{\{0,1,3\}\}$.
  • Figure 2: Flags are drawn in the left figure and represented by little triangles around hypervertices in the right figure.
  • Figure 3: Two orbits of flags are represented by black little triangles and white little triangles.
  • Figure 4: Three figures show the orbit of $\langle\gamma_1\gamma_2\rangle$, $\langle\gamma_0\gamma_2\rangle$ and $\langle\gamma_0\gamma_1\rangle$ on $F_1$.
  • Figure 5: $\hbox{\rm CH}(\mathbb{Z}_7,Y)$ with $Y=\{\{0,1,3\}\}$ embedded on Torus (the left figure) and the orientable surface of genus $3$ (the right figure).

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 27 more