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Regular embedding of simple hypergraphs

Yanhong Zhu, Kai Yuan

TL;DR

This work studies regular embeddings of simple hypergraphs via their Levi graphs and associated hypermaps, focusing on high-symmetry regular hypermaps. It develops an algorithm to classify regular embeddings of simple hypergraphs for a given order and determines automorphism groups for prime square orders by analyzing transitive permutation groups with dihedral point stabilizers. The authors establish a key edge-multiplicity criterion tied to the core of stabilizers, classify primitive and imprimitive affine-type groups for degree $p^2$, and enumerate explicit automorphism groups $G_i$ (with $G_7$ excluded) alongside concrete regular simple hypermaps for order $p^2$. The results provide a concrete, computable framework for enumerating regular embeddings of simple hypergraphs of prime-square order and mapping their symmetry groups to explicit group presentations.

Abstract

Regular hypermaps with underlying simple hypergraphs are analysed. We obtain an algorithm to classify the regular embeddings of simple hypergraphs with given order, and determine the automorphism groups of regular embedding of simple hypergraphs with prime square order.

Regular embedding of simple hypergraphs

TL;DR

This work studies regular embeddings of simple hypergraphs via their Levi graphs and associated hypermaps, focusing on high-symmetry regular hypermaps. It develops an algorithm to classify regular embeddings of simple hypergraphs for a given order and determines automorphism groups for prime square orders by analyzing transitive permutation groups with dihedral point stabilizers. The authors establish a key edge-multiplicity criterion tied to the core of stabilizers, classify primitive and imprimitive affine-type groups for degree , and enumerate explicit automorphism groups (with excluded) alongside concrete regular simple hypermaps for order . The results provide a concrete, computable framework for enumerating regular embeddings of simple hypergraphs of prime-square order and mapping their symmetry groups to explicit group presentations.

Abstract

Regular hypermaps with underlying simple hypergraphs are analysed. We obtain an algorithm to classify the regular embeddings of simple hypergraphs with given order, and determine the automorphism groups of regular embedding of simple hypergraphs with prime square order.
Paper Structure (9 sections, 21 theorems, 20 equations)

This paper contains 9 sections, 21 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Embedding of simple hypergraphs
  3. Hypergraphs
  4. Hypermaps
  5. Some Results
  6. Transitive Permutation Groups
  7. $G$ is a primitive group
  8. $G$ is an imprimitive group
  9. Regular Hypermaps of Order Prime Square

Key Result

Proposition 2.4

Let $\mathcal{H} = \mathcal{H}(G; \gamma_0, \gamma_1, \gamma_2)$ be a regular hypermap with simple underlying Levi graph. Then $\langle \gamma_1,\gamma_2\rangle \cap \langle\gamma_0,\gamma_2\rangle=\langle\gamma_2\rangle$.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Corollary 2.8
  • ...and 26 more