Hamiltonicity of Transitive Graphs Whose Automorphism Group Has $\Z_{p}$ as Commutator Subgroups

TL;DR

This paper extends a 1982 result by proving Hamiltonicity in certain infinite transitive graphs with automorphism groups having a cyclic prime order commutator subgroup, broadening the scope from finite to infinite graphs.

Contribution

It generalizes a known finite group Hamiltonicity result to infinite graphs with specific automorphism group properties, introducing new classes of Hamiltonian graphs.

Findings

Hamiltonian cycles exist in the specified infinite graphs.

Generalization from finite to infinite graph cases.

Broader class of graphs with cyclic prime order commutator subgroups.

Abstract

In 1982, Durnberger proved that every connected Cayley graph of a finite group with a commutator subgroup of prime order contains a hamiltonian cycle. In this paper, we extend this result to the infinite case. Additionally, we generalize this result to a broader class of infinite graphs $X$, where the automorphism group of $X$ contains a transitive subgroup $G$ with a cyclic commutator subgroup of prime order.