bibkey: kumar2026weakmetric
authors: Aryan Kumar
year: 2026
title: Weak k-metric dimension of Hamming graphs: rectangular products and near-maximum parameters
doi: 10.48550/arXiv.2609.32161
url: https://arxiv.org/abs/2609.32161
claim: "Section 6 proves an eventual near-maximum formula with an explicit sufficient cutoff; Section 7 asks for the sharp nonbinary threshold."
strata_touched:
  - D5/S3/Combinatorics/Hamming/TernaryWeakMetricStability
license: citation-only
triage: anchor

Near-maximum weak metric dimension of Hamming graphs

The source is arXiv:2609.32161v1, 26 September 2026. The abstract and current HTML were checked on 28 October 2026; the submission history lists v1.

A weak k-resolving set S satisfies sum over w in S of |rho(x,w)-rho(y,w)| >= k for every distinct x,y. Here rho is Hamming distance, not the number of distinguishing landmarks. For q >= 3, put kappa = 2 q^(d-1) and U_q(t) = q floor(t/2) + (t mod 2).

Lemma 6.3 gives the adjacent-coordinate frequency restriction and the bound |T| <= U_q(t) for omitted sets T. Theorem 6.4 proves eventual equality wdim_(kappa-t)(K_q square d) = q^d - U_q(t), under kappa > t, d >= 2, and 2(q-2)q^(d-2) >= d U_q(t)-t. Its construction uses simultaneous alphabet translation orbits and an additional vertex for odd t.

Section 7 says that determining a sharp threshold for the nonbinary formula in Theorem 6.4 is another natural question. The retained trureturing target only concerns q=3 and t in {4,5}; it does not settle that whole nonbinary question. The adjacent bound and orbit idea are attributed prerequisites. The proposed improvement is the all-pair sufficiency estimate and the exact least threshold for those two deficits. No global priority claim is made.

Verified locator

DOI: https://doi.org/10.48550/arXiv.2609.32161

Source: https://arxiv.org/abs/2609.32161

Lemma 6.3, Theorem 6.4 and Section 7 locate the adjacent-frequency bound, eventual near-maximum formula and sharp-threshold question described above.