Weak k-metric dimension of Hamming graphs: rectangular products and near-maximum parameters
Abstract
We determine weak k-metric dimensions for several families of Hamming graphs. For n≥3, m>n, and 3≤ k≤2n, a cyclic construction proves Conjecture 6.1 of Fernández, Klavžar, Kuziak, Muñoz-Márquez and Yero (2026) on Kn□ Km. For these rectangular products, we also prove that wdim2(Kn□ Km)=m exactly when m≥2n−2. For hypercubes Qd with d≥2, we show that consecutive parameters 2s−1 and 2s have identical weak resolving sets. Near the maximum parameter, a reduction to restricted-distance binary codes determines wdim2d−t(Qd) for every feasible deficit 0≤ t≤15. For L≥1 and t∈{2L,2L+1}, we prove the stabilization formula wdim2d−t(Qd)=2d−L for d≥2L+1, and show that this threshold is sharp. For each fixed q≥3 and deficit t, we also determine wdim2qd−1−t(Kq□ d) in all sufficiently large dimensions, with an explicit sufficient condition.
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BibTeX
@misc{kumar2026weak,
title = {{Weak $k$-metric dimension of Hamming graphs: rectangular products and near-maximum parameters}},
author = {Aryan Kumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32161},
eprint = {2609.32161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32161},
}
RIS
TY - UNPB
TI - Weak k-metric dimension of Hamming graphs: rectangular products and near-maximum parameters
AU - Aryan Kumar
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32161
UR - https://arxiv.org/abs/2609.32161
AB - We determine weak k-metric dimensions for several families of Hamming graphs. For n≥3, m>n, and 3≤ k≤2n, a cyclic construction proves Conjecture 6.1 of Fernández, Klavžar, Kuziak, Muñoz-Márquez and Yero (2026) on K_n□ K_m. For these rectangular products, we also prove that wdim_2(K_n□ K_m)=m exactly when m≥2n−2. For hypercubes Q_d with d≥2, we show that consecutive parameters 2s−1 and 2s have identical weak resolving sets. Near the maximum parameter, a reduction to restricted-distance binary codes determines wdim_2^d−t(Q_d) for every feasible deficit 0≤ t≤15. For L≥1 and t∈{2L,2L+1}, we prove the stabilization formula wdim_2^d−t(Q_d)=2^d−L for d≥2^L+1, and show that this threshold is sharp. For each fixed q≥3 and deficit t, we also determine wdim_2q^d−1−t(K_q^□ d) in all sufficiently large dimensions, with an explicit sufficient condition.
ER -