bibkey: farhan2026weakdirect
authors: Mohammad Farhan, Dorota Kuziak, Ismael G. Yero
year: 2026
title: "The weak k-metric dimension of the direct product of complete graphs"
doi: 10.48550/arXiv.2605.22307
url: https://arxiv.org/html/2605.22307v1
claim: "Section 3 paragraph S3.p17.1 asks for the exact weak-four metric dimension of K_n times K_n for every n at least four; Section 6 first bullet asks whether Theorem 3.3 is sharp."
strata_touched:
  - D5/S3/Combinatorics/Graph/DirectSquareWeakFour/Exact
license: citation-only
triage: anchor

Direct-square weak-four metric dimension

Verified locator

DOI: 10.48550/arXiv.2605.22307

URL: https://arxiv.org/html/2605.22307v1

Version: arXiv:2605.22307v1. Proposition 2.2 gives the distinct-pair coordinate identities. Section 3, paragraph S3.p17.1, asks for the exact weak-four value at every order at least four. Section 6, first bullet, identifies it as the remaining exact-value case and asks about the sharpness of Theorem 3.3’s upper bound.

Source question

The result above clearly leads to a question concerning the exact value of wdim4(Kn × Kn) when n ≥ 4.

The product is direct: both coordinates differ along an edge. The introductory weak-resolution definition omits the distinct-pair restriction, which is explicit in Proposition 2.2. Self-pairs have zero separation, so the stated positive-parameter results require that restriction.

Repository result and boundary

The owning theorem proves the exact value $2n+\lceil 2n/11\rceil +1_{n=5}$ for every $n\ge 4$, using actual shortest graph distances, a universal lower bound, an order-five obstruction, and equality-size constructions at every order. The source is a problem input, not an assumed theorem establishing this answer.

The question and source version are identified exactly. Bounded prior searches did not find the displayed answer in their inspected windows. Complete forward citations, uninspected cited bodies, non-arXiv literature and worldwide priority remain unverified.