bibkey: bajnokbeatty2025diameter
authors: Bela Bajnok and W. Kyle Beatty
year: 2025
title: On the diameter of undirected Cayley graphs of finite abelian groups
doi: 10.2140/cnt.2025.14.1
url: https://arxiv.org/abs/2406.04045v1
claim: Conjecture 4.1 concerns signed two-generator diameter in Z/2 times Z/(2k); the formal result proves one uniform sufficient family and leaves the converse open.
strata_touched:
  - D5/S3/FiniteGroups/BajnokBeattyCayleySufficiency
license: citation-only
triage: anchor

On the diameter of undirected Cayley graphs of finite abelian groups

Bela Bajnok and W. Kyle Beatty, Combinatorics and Number Theory 14 (2025), no. 1, 1–12. The accessible full text checked here is arXiv:2406.04045v1, submitted 6 June 2024. The publisher records a revision on 26 October 2024; the published full text was not accessed because the download returned subscription-access-denied HTML. Conclusions about the paper’s precise theorem scope refer to arXiv v1, and do not certify literature novelty.

In Section 4, an s-spanning pair represents every group element with integer coefficients whose absolute values sum to at most s. Conjecture 4.1 gives a classification for Z/2 × Z/(2k). Theorem 4.2 proves a necessary order bound; Theorem 4.3 treats odd s; Theorem 4.4 gives sufficiency for even s only when k ≤ (s²−s)/2. Theorem 1.9 asserts the maximal order 2s² at even radius for groups of this form; it therefore already attests the j=0 endpoint. That endpoint is not claimed as a new positive case here.

The earlier positive constructions do not directly supply the whole retained family. Proposition 2.2 is cyclic. Proposition 2.3 concerns Z/s × Z/(2s). Theorem 2.4 requires s mod c ≥ (c−1)/2, which for c=2 allows only odd s. Corollary 2.5 has odd first invariant factor.

The repository construction proves: for t ≥ 1 and 0 ≤ j < t with t even or j even, k=2(t²−j) admits a distinct signed spanning pair of radius at most 2t. It therefore includes the allowed top strip 0 ≤ 2j < t. A simultaneous integer remainder descent supplies the covering, and explicit surjective maps supply the exact group Z/2 × Z/(2k). This is a repo-derived sufficient construction. It proves neither exclusion of other pairs nor the conjecture’s general necessity direction. The nonprimitive excluded displayed lattice does not exclude alternative primitive lattices.

Verified locator

DOI: https://doi.org/10.2140/cnt.2025.14.1

Source: https://arxiv.org/abs/2406.04045v1

Necessary congruences in the even-radius top strip

The repository’s necessity result applies to every pair in Z/2 × Z/(2k). For integers t ≥ 1 and 0 ≤ r < t with k=2t²−r, a signed coefficient cover of radius 2t forces 2 | r and, when t is odd, 4 | r. It imposes no chosen lattice form or distinctness hypothesis on the pair.

The mechanism uses the actual relation quotient. For n ≥ 2 and 0 ≤ δ < n, a square [0,n)² covering Z²/L with quotient order n²−δ forces L to have basis (n,−εb), (−a,εn), where ε=±1, 0 ≤ a,b < n and ab=δ. Least adjacent-column displacements and determinant divisibility force this basis; the determinant-index identity is supplied by pinned Mathlib, not claimed as new mathematics.

For an arbitrary covering pair, the affine odd coefficient class becomes a square torsor under the sum-and-difference map. Its relation quotient has order 2k, so the rigidity result applies with n=2t and δ=2r. Parity forces both displacement parameters to be even. When t is odd, failure of 4 | r would force the entire original relation kernel into 4Z², contradicting the first Z/2 factor. Thus the argument excludes alternative pairs, beyond checking the displayed sufficient construction.

This is a repo-derived necessary direction in the stated top strip, alongside the sufficient family above. The source comparison remains arXiv:2406.04045v1, Conjecture 4.1 and Theorems 4.2–4.4. The revised paper has not been read; neither the full conjecture nor literature novelty is asserted by this increment.