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 $n\ge3$, $m>n$, and $3\le k\le2n$, a cyclic construction proves Conjecture 6.1 of Fern\'andez, Klav\v{z}ar, Kuziak, Mu\~noz-M\'arquez and Yero (2026) on $K_n\square K_m$. For these rectangular products, we also prove that $\operatorname{wdim}_2(K_n\square K_m)=m$ exactly when $m\ge2n-2$. For hypercubes $Q_d$ with $d\ge2$, 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 $\operatorname{wdim}_{2^d-t}(Q_d)$ for every feasible deficit $0\le t\le15$. For $L\ge1$ and $t\in\{2L,2L+1\}$, we prove the stabilization formula $\operatorname{wdim}_{2^d-t}(Q_d)=2^d-L$ for $d\ge2^L+1$, and show that this threshold is sharp. For each fixed $q\ge3$ and deficit $t$, we also determine $\operatorname{wdim}_{2q^{d-1}-t}(K_q^{\square d})$ in all sufficiently large dimensions, with an explicit sufficient condition.