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Hamiltonicity in generalized quasi-dihedral groups

Babak Miraftab, Konstantinos Stavropoulos

TL;DR

The paper proves that every connected Cayley graph of a two-ended generalized quasi-dihedral group $G=A\ast_K B$ contains a Hamiltonian double ray, extending the finite-case result for (generalized) dihedral groups to the infinite setting. The authors develop a constructive framework by extracting spanning grid- and cylinder-like subgraphs (including twisted cubic cylinders $\overline{W}_{k,l}$) from Cayley graphs and proving these subgraphs admit Hamiltonian double rays or circles; they then apply a generator-size induction to lift the local Hamiltonicity of these structures to the entire Cayley graph. Key technical tools include width-two reductions, 6- and 4-cycles in two-ended GQD Cayley graphs, and the systematic use of grids to assemble spanning double rays. The results advance understanding of Hamiltonian properties in infinite, two-ended, vertex-transitive graphs and motivate further exploration of Hamiltonian circles, Hamilton-connectivity, and decomposition questions within this class.

Abstract

Witte Morris showed in [21] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product with amalgamation $\mathbb Z_2 \ast \mathbb Z_2$. We show that every connected Cayley graph of the infinite dihedral group has both a Hamiltonian double ray, and extend this result to all two-ended generalized quasi-dihedral groups.

Hamiltonicity in generalized quasi-dihedral groups

TL;DR

The paper proves that every connected Cayley graph of a two-ended generalized quasi-dihedral group contains a Hamiltonian double ray, extending the finite-case result for (generalized) dihedral groups to the infinite setting. The authors develop a constructive framework by extracting spanning grid- and cylinder-like subgraphs (including twisted cubic cylinders ) from Cayley graphs and proving these subgraphs admit Hamiltonian double rays or circles; they then apply a generator-size induction to lift the local Hamiltonicity of these structures to the entire Cayley graph. Key technical tools include width-two reductions, 6- and 4-cycles in two-ended GQD Cayley graphs, and the systematic use of grids to assemble spanning double rays. The results advance understanding of Hamiltonian properties in infinite, two-ended, vertex-transitive graphs and motivate further exploration of Hamiltonian circles, Hamilton-connectivity, and decomposition questions within this class.

Abstract

Witte Morris showed in [21] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product with amalgamation . We show that every connected Cayley graph of the infinite dihedral group has both a Hamiltonian double ray, and extend this result to all two-ended generalized quasi-dihedral groups.
Paper Structure (10 sections, 29 theorems, 29 equations, 5 figures)

This paper contains 10 sections, 29 theorems, 29 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of GQD groups
  4. Short cycles in Cayley graphs of two-ended GQD groups
  5. Grids, Walls and Cylinders
  6. Proof of the main Theorem
  7. Final Remarks
  8. Hamiltonian circles
  9. Hamilton-connectivity
  10. Decomposition and Uniqueness

Key Result

Theorem 1.1

Let $G=A\ast_K A$, where $A$ is a generalized quasi-dihedral on $K$. Then every connected Cayley graph of $G$ contains a Hamiltonian double ray.

Figures (5)

  • Figure 1: The gray path in the left picture depicts $S_{0,4}$ in $\overline{W}_{4,4}$ and the gray path in the right picture depicts $S_{0,4}$ in $\overline{W}_{5,3}$.
  • Figure 2: A column (blue) and a staircase (red) in the twisted cubic cylinder $\overline{W}_{4,4}$.
  • Figure 3: The double rays of $\overline{W}_{4,2}$ that serve as rows of $\overline{W}_{3,5}$. The red wavy edges correspond to twisted edges between the first(blue) and last(green) double ray of $\overline{W}_{3,5}$.
  • Figure 4: The union of the blue and red double rays covers all vertices of $\overline{W}_{3,3}$
  • Figure 5: The partition of $G$ according to the cosets of $H'$.

Theorems & Definitions (68)

  • Conjecture 1: miraftab2017cycles
  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Example
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5: cf. RDsBanffSurveyI
  • ...and 58 more