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arxiv: 2304.03348 · v2 · submitted 2023-04-06 · 🧮 math.CO

Cayley graphs of order 8pq are hamiltonian

Fateme Abedi , Dave Witte Morris , Javanshir Rezaee , M. Reza Salarian This is my paper

Pith reviewed 2026-05-24 09:29 UTC · model grok-4.3

classification 🧮 math.CO
keywords Cayley graphsHamiltonian cyclesgroups of order 8pqcomputer-assisted proofsfinite groupsgraph theory
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The pith

If G has order 8pq for distinct primes p and q then every connected Cayley graph on G has a Hamiltonian cycle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper uses computer assistance to prove that for any finite group G of order 8pq with p and q distinct primes, every connected Cayley graph on G contains a Hamiltonian cycle. This verifies a special case of the conjecture that connected Cayley graphs are Hamiltonian. A reader might care because it confirms the property for all groups in this order class, including both abelian and nonabelian ones. The proof works by exhaustively listing the groups and checking their graphs.

Core claim

If G is a finite group of order 8pq, where p and q are distinct primes, then every connected Cayley graph on G has a hamiltonian cycle. This is established via computer-assisted enumeration and verification.

What carries the argument

Computer-assisted enumeration and checking of all groups of order 8pq and their connected Cayley graphs for the presence of Hamiltonian cycles.

Load-bearing premise

The computer verification correctly enumerates all groups of order 8pq, all possible connection sets, and all connected Cayley graphs without implementation or classification errors.

What would settle it

A single counterexample group of order 8pq with a connected Cayley graph lacking a Hamiltonian cycle would disprove the claim.

read the original abstract

We give a computer-assisted proof that if $G$ is a finite group of order $8pq$, where $p$ and $q$ are distinct primes, then every connected Cayley graph on $G$ has a hamiltonian cycle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to give a computer-assisted proof that every connected Cayley graph on a finite group G of order 8pq (p, q distinct primes) admits a Hamiltonian cycle, via exhaustive classification of all such groups, all possible connection sets yielding connected Cayley graphs, and direct verification of Hamiltonicity in each case.

Significance. If the computational enumeration and verification are correct, the result would confirm the Hamiltonian conjecture for Cayley graphs in this order class, providing a complete case analysis for |G|=8pq. The exhaustive approach is a strength when accompanied by reproducible code or explicit case lists, as it avoids reliance on unproven reductions.

major comments (2)
  1. [Computational methods / case analysis] The central claim rests entirely on computer-assisted enumeration of groups of order 8pq and verification that each connected Cayley graph is Hamiltonian, yet the manuscript supplies neither source code, an explicit list of checked cases, nor a description of the Hamiltonicity algorithm (e.g., backtracking or reduction steps). This is load-bearing, as any omission in group classification or connection-set generation would falsify the claim.
  2. [Group classification] The group classification step must correctly enumerate all abelian and non-abelian groups of order 8pq for every pair of distinct primes p,q (including cases where p≡1 mod 8 or q≡1 mod 4). The manuscript should state the total number of groups per congruence class and confirm agreement with independent sources such as the SmallGroups library or known results on groups of order 8p and 8q.
minor comments (1)
  1. [Introduction / preliminaries] Notation for connection sets S and the Cayley graph Cay(G,S) should be introduced once and used consistently; currently the transition between abstract group theory and computational implementation is abrupt.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive comments on our manuscript. We address each major comment below and indicate the revisions planned.

read point-by-point responses

  1. Referee: [Computational methods / case analysis] The central claim rests entirely on computer-assisted enumeration of groups of order 8pq and verification that each connected Cayley graph is Hamiltonian, yet the manuscript supplies neither source code, an explicit list of checked cases, nor a description of the Hamiltonicity algorithm (e.g., backtracking or reduction steps). This is load-bearing, as any omission in group classification or connection-set generation would falsify the claim.

    Authors: We agree that the current version lacks sufficient detail on the computational verification for full reproducibility. The revised manuscript will include an explicit description of the Hamiltonicity algorithm (including the backtracking procedure and any reductions used), a summary table listing the number of groups and connection sets verified for representative classes of p and q, and a link to a public repository containing the source code. revision: yes

  2. Referee: [Group classification] The group classification step must correctly enumerate all abelian and non-abelian groups of order 8pq for every pair of distinct primes p,q (including cases where p≡1 mod 8 or q≡1 mod 4). The manuscript should state the total number of groups per congruence class and confirm agreement with independent sources such as the SmallGroups library or known results on groups of order 8p and 8q.

    Authors: We will revise the manuscript to state the total number of groups (abelian and non-abelian) for each congruence class of the primes p and q. A new subsection will confirm that the enumeration matches the SmallGroups library in GAP and known classifications for groups of order 8p and 8q, with explicit counts provided for the relevant cases. revision: yes

Circularity Check

0 steps flagged

No circularity; direct computer-assisted classification proof.

full rationale

The paper claims a computer-assisted exhaustive proof that all connected Cayley graphs on groups of order 8pq are Hamiltonian, via classification of groups, connection sets, and direct verification. No equations, definitions, or steps reduce by construction to fitted inputs or self-citations. The derivation is self-contained as an enumeration rather than a self-referential or renamed result. Absence of published code affects verifiability but does not create circularity under the specified criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions of groups, Cayley graphs, and Hamiltonian cycles together with the unverified correctness of the computer enumeration and search procedures.

axioms (1)
  • standard math Standard axioms of finite group theory and graph theory
    The proof invokes the usual definitions of Cayley graphs, connectedness, and cycles without introducing new axioms.

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Reference graph

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