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On automorphism groups of half-arc-transitive tetravalent graphs

Yuandong Li, Binzhou Xia, Jin-Xin Zhou

TL;DR

This work completes a structural classification of tetravalent graphs Γ admitting a maximal $(\frac{1}{2},t)$-pair $(M,H)$ with $M<H$, under which Γ is $M$-half-arc-transitive and $H$-arc-transitive. It proves that for $t\ge2$ only two quotient-type configurations can occur ($t=2$ or $t=3$) and analyzes the $t=1$ case to reveal either a normal inclusion or a specific non-normal action with a semiregular kernel, and it demonstrates that all cases admit explicit examples. Among the contributions are the first known tetravalent normal-edge-transitive non-normal Cayley graph on a nonabelian simple group and a counterexample to a question of Rivera–Šparl regarding alternating-cycle graphs. The results advance understanding of automorphism groups of tetravalent G-HAT graphs and the structure of their alternating-cycle quotients, with computational verification via Magma supporting the constructions.

Abstract

We characterize connected tetravalent graphs $Γ$ which admit groups $M<H$ of automorphisms such that $Γ$ is $M$-half-arc-transitive and $H$-arc-transitive. Examples for each case are constructed, including a counter-example to a question asked by A. R. Rivera and P. Šparl in 2019 as well as the first example of tetravalent normal-edge-transitive non-normal Cayley graph on a nonabelian simple group.

On automorphism groups of half-arc-transitive tetravalent graphs

TL;DR

This work completes a structural classification of tetravalent graphs Γ admitting a maximal -pair with , under which Γ is -half-arc-transitive and -arc-transitive. It proves that for only two quotient-type configurations can occur ( or ) and analyzes the case to reveal either a normal inclusion or a specific non-normal action with a semiregular kernel, and it demonstrates that all cases admit explicit examples. Among the contributions are the first known tetravalent normal-edge-transitive non-normal Cayley graph on a nonabelian simple group and a counterexample to a question of Rivera–Šparl regarding alternating-cycle graphs. The results advance understanding of automorphism groups of tetravalent G-HAT graphs and the structure of their alternating-cycle quotients, with computational verification via Magma supporting the constructions.

Abstract

We characterize connected tetravalent graphs which admit groups of automorphisms such that is -half-arc-transitive and -arc-transitive. Examples for each case are constructed, including a counter-example to a question asked by A. R. Rivera and P. Šparl in 2019 as well as the first example of tetravalent normal-edge-transitive non-normal Cayley graph on a nonabelian simple group.
Paper Structure (5 sections, 10 theorems, 10 equations, 1 table)

This paper contains 5 sections, 10 theorems, 10 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterization of maximal $(\frac{1}{2}, t)$-pairs
  4. Examples
  5. Proof of main results

Key Result

Theorem 1.3

Let $\mathrm{\Gamma}$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\mathrm{\Gamma}$ with $t\geq1$, and let $K=\mathrm{Core}_H(M)$. Take $u\in V(\mathrm{\Gamma})$. Then one of the followings holds. Moreover, examples for each of the above cases exist.

Theorems & Definitions (17)

  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1: AAMPS2016
  • Lemma 2.2: Praeger1993
  • Lemma 2.3
  • Lemma 2.4: DPZ
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 7 more