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Basic Tetravalent Oriented Graphs with Cyclic Normal Quotients

Nemanja Poznanovic, Cheryl E. Praeger

TL;DR

This work completes a program to classify basic pairs in $\mathcal{OG}(4)$ by focusing on cycle-type basic pairs with nondegenerate cyclic normal quotients. It develops a cyclic-quotient framework distinguishing oriented-cycle, unoriented-cycle, and independent-cycle types, and derives sharp bounds on the socle structure $\mathrm{soc}(G)=T^k$ (with $T$ nonabelian or abelian) in terms of the largest cyclic quotient length $r$; for example, $k\le r2^r$ in oriented nonabelian cases and $k\le 2r$ or $r-1$ in unoriented cases, depending on the socle type. The authors provide extensive constructions—via coset graphs and explicit generating pairs—to realize infinite families attaining these bounds, including examples with abelian and nonabelian socles and both oriented and unoriented cyclic quotients; notably, a final construction yields $k=24$ when $r=3$, confirming tightness in at least one regime. Collectively, the results deliver a complete description of basic cycle-type pairs and demonstrate how all members of $\mathcal{OG}(4)$ arise as normal covers of such basic cycle-type graphs, advancing understanding of symmetry, normal quotients, and cycle quotients in tetravalent $G$-oriented graphs.

Abstract

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(Γ, G)$ where $Γ$ is finite, 4-valent, connected, and $G$-oriented ($G$-half-arc-transitive). A subfamily of $\mathcal{OG}(4)$ has recently been identified as `basic' in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one $G$-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in $\mathcal{OG}(4)$.

Basic Tetravalent Oriented Graphs with Cyclic Normal Quotients

TL;DR

This work completes a program to classify basic pairs in by focusing on cycle-type basic pairs with nondegenerate cyclic normal quotients. It develops a cyclic-quotient framework distinguishing oriented-cycle, unoriented-cycle, and independent-cycle types, and derives sharp bounds on the socle structure (with nonabelian or abelian) in terms of the largest cyclic quotient length ; for example, in oriented nonabelian cases and or in unoriented cases, depending on the socle type. The authors provide extensive constructions—via coset graphs and explicit generating pairs—to realize infinite families attaining these bounds, including examples with abelian and nonabelian socles and both oriented and unoriented cyclic quotients; notably, a final construction yields when , confirming tightness in at least one regime. Collectively, the results deliver a complete description of basic cycle-type pairs and demonstrate how all members of arise as normal covers of such basic cycle-type graphs, advancing understanding of symmetry, normal quotients, and cycle quotients in tetravalent -oriented graphs.

Abstract

Let denote the family of all graph-group pairs where is finite, 4-valent, connected, and -oriented (-half-arc-transitive). A subfamily of has recently been identified as `basic' in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one -normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in .
Paper Structure (21 sections, 34 theorems, 54 equations, 1 figure, 4 tables)

This paper contains 21 sections, 34 theorems, 54 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. G-Oriented Graphs
  4. Vertex Stabilisers in $\mathop{\mathrm{\mathcal{OG}}}\nolimits(4)$
  5. Normal Quotients in $\mathop{\mathrm{\mathcal{OG}}}\nolimits(4)$
  6. Cyclic Normal Quotients
  7. Describing the Socle
  8. Oriented-Cycle Type
  9. Oriented-Cycle Type: nonabelian socle
  10. Oriented-Cycle Type: abelian socle
  11. Unoriented-Cycle Type
  12. Unoriented-Cycle Type: nonabelian socle
  13. Unoriented-Cycle Type: abelian socle
  14. Proof of Theorem \ref{['CycleMainTheorem']}
  15. Constructions of basic pairs
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose $(\Gamma, G) \in \mathop{\mathrm{\mathcal{OG}}}\nolimits(4)$ is basic of cycle type and does not have independent cyclic normal quotients. Then one of the following holds.

Figures (1)

  • Figure 1: Subgroup lattice showing structure of $G$, depending on whether $\Gamma_N$ is $G$-oriented or $G$-unoriented.

Theorems & Definitions (65)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 1
  • Remark 4.1
  • Lemma 4.2
  • ...and 55 more