On minimal graphs for hamiltonian groups and their fixing set

TL;DR

This work addresses the problem of realizing finite hamiltonian groups as automorphism groups of graphs and determining both the minimal vertex order $\alpha(H)$ and the fixing set $\mathrm{fix}(H)$ for such groups $H \cong Q_8 \times A$. It establishes $\alpha(H) = 16 + \alpha(A)$ when $A$ contains no element of order $4$, yielding the explicit $\alpha(Q_8 \times C_2^{k}) = 16 + 2k$, and shows $\alpha(H) \ge \mu(H)$ with matching upper bounds in key cases. On the fixing side, the paper proves $\mathrm{fix}(H) = \{1,2,\dots, d+1\}$ for a periodic abelian $A$ with no element of order $4$, where $d$ is the number of elementary divisors of $A$, by combining a known result $\mathrm{fix}(Q_8) = \{1\}$ with a product-structure lemma. These results partially address an open problem about fixing sets under direct products and provide a framework for analyzing symmetry-breaking in graphs with hamiltonian symmetry, linking minimal graphs, automorphism groups, and fixing parameters in a concrete, computable way.

Abstract

A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. The fixing number of a graph $Γ$ is the minimum cardinality of a subset $S$ of $V(Γ)$ such that the stabilizer of $S$ is trivial. For a given finite group $G$, the fixing set is defined as the set comprising all possible fixing numbers of graphs having group $G$ as their automorphism groups. We determine the fixing sets corresponding to finite hamiltonian groups.

Figures (2)

• Figure 1: Vertex-minimal graphs with $\mathop{\mathrm{Aut}} \Gamma_{1} = Q_{8}$ and $\mathop{\mathrm{Aut}} \Gamma_{2} = C_{2}^{4}$.
• Figure 2: Graphs with identity automorphism

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof