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 $n\ge 3$, $m>n$, and $3\le k\le 2n$, a cyclic construction proves Conjecture 6.1 of Fernández, Klavžar, Kuziak, Muñoz-Márquez and Yero (2026) on $K_n\square K_m$. For these rectangular products, we also prove that ${\mathrm{wdim}}_2(K_n\square K_m)=m$ exactly when $m\ge 2n-2$. For hypercubes $Q_d$ with $d\ge 2$, 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 ${\mathrm{wdim}}_{2^d-t}(Q_d)$ for every feasible deficit $0\le t\le 15$. For $L\ge 1$ and $t\in \{2L,2L+1\}$, we prove the stabilization formula ${\mathrm{wdim}}_{2^d-t}(Q_d)=2^d-L$ for $d\ge 2^L+1$, and show that this threshold is sharp. For each fixed $q\ge 3$ and deficit $t$, we also determine ${\mathrm{wdim}}_{2q^{d-1}-t}(K_q^{\square d})$ in all sufficiently large dimensions, with an explicit sufficient condition.

arXiv comment21 pages; ancillary files for two finite capacity verifications