[Paper Review] 2-generated Cayley digraphs on nilpotent groups have hamiltonian paths

Dave Witte Morris|arXiv (Cornell University)|Mar 28, 2011

graph theory and CDMA systems2 references4 citations

TL;DR

This paper proves that every 2-generated Cayley digraph on a finite nilpotent group admits a Hamiltonian path, extending known results for abelian and p-groups. Using subgroup induction and properties of arc-forcing subgroups, the authors establish that such graphs—especially those of valence ≤4—always contain a Hamiltonian path, resolving a key case in the broader conjecture on Hamiltonian paths in Cayley graphs on non-abelian groups.

ABSTRACT

Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies there is a hamiltonian path in every connected Cayley graph on G that has valence at most 4.

Motivation & Objective

• To determine whether all connected Cayley digraphs on nilpotent groups admit Hamiltonian paths, extending known results for abelian and p-groups.
• To investigate whether the nilpotency condition is sufficient to guarantee Hamiltonian paths in Cayley digraphs, especially when the group is not abelian or of prime power order.
• To establish that all connected Cayley graphs of valence ≤4 on nilpotent groups have Hamiltonian paths, addressing a long-standing open problem in group theory.
• To provide a structural proof using subgroup series and arc-forcing subgroups, generalizing prior results on p-groups and abelian groups.
• To demonstrate that the word 'nilpotent' cannot be weakened to 'solvable' or 'supersolvable', as counterexamples exist in those classes.

Proposed method

• Uses induction on the group order, leveraging the structure of nilpotent groups as direct products of p-groups.
• Applies the concept of the arc-forcing subgroup $ H = \\<S^{-1}S\rangle $, which is shown to be abelian in 2-generated cases, enabling application of known results on abelian groups.
• Employs a subnormal series $ H = H_1 \triangleleft \cdots \triangleleft H_m = H^G $ with prime-power factor groups, exploiting nilpotency to build Hamiltonian paths step-by-step.
• Relies on the fact that every connected Cayley digraph on an abelian group or a p-group has a Hamiltonian path, as established in prior work.
• Uses the lifting technique: given Hamiltonian paths in quotient groups and subgroups, constructs a Hamiltonian path in the full group via concatenation and coset traversal.
• Applies a generalization of the p-group result (Theorem 2.2) to normal p-subgroups, ensuring Hamiltonian cycles in relevant quotient structures.

Experimental results

Research questions

• RQ1 Do all 2-generated Cayley digraphs on finite nilpotent groups have Hamiltonian paths?
• RQ2 Can the condition of nilpotency be extended to cover all connected Cayley graphs of valence ≤4 on such groups?
• RQ3 Is the nilpotency assumption necessary, or can it be weakened to solvability or supersolvability?
• RQ4 What structural properties of the arc-forcing subgroup $ \langle S^{-1}S \rangle $ ensure the existence of Hamiltonian paths?
• RQ5 How do subgroup series and quotient group structures facilitate the construction of Hamiltonian paths in nilpotent groups?

Key findings

• Every 2-generated Cayley digraph on a finite nilpotent group has a Hamiltonian path, as the arc-forcing subgroup $ \langle a^{-1}b \rangle $ is cyclic and hence abelian.
• All connected Cayley graphs of valence ≤4 on nilpotent groups have Hamiltonian paths, since such graphs correspond to generating sets of size at most 2 with no involutions.
• The result holds even when the group is a direct product $ P \times A $ with $ P $ a p-group and $ A $ abelian, generalizing the p-group and abelian cases.
• The proof shows that the arc-forcing subgroup being abelian is sufficient for Hamiltonian path existence in the Cayley digraph, under nilpotency.
• Counterexamples exist for solvable or supersolvable groups, showing that nilpotency is a necessary condition for the generalization to hold.
• A generalization of the p-group result is established: if $ S \subset aN $ for a normal p-subgroup $ N $, then $ \overrightarrow{\operatorname{Cay}}(G;S) $ has a Hamiltonian cycle.