AI 中文总结
该研究解决了图的多重集维数领域的一个猜想和两个开放问题，确定了达到多重集维数平凡上界的最小图阶数，证明了正方形国王网格的多重集维数为4，还得出了国王带多重集维数的线性增长规律。
AI 中文摘要
我们给出图的多重集维数的三个结果，解决了文献中的一个猜想和两个开放问题。首先，我们反驳了Simanjuntak、Siagian和Vetrík（2017）提出的“所有阶数为$n(G)$且具有有限多重集维数的图$G$满足$\text{dim}_m(G) \le n(G)-1$”的猜想：对阶数2至11的共1018690328个连通图进行穷举计算后发现，恰好有8个图达到$\text{dim}_m(G)=n(G)$，且全部为阶数11的图，因此阶数11是达到平凡上界的最小阶数。这也回答了Farhan、Klavžar、Kuziak和Yero近期综述中的一个问题。其次，我们证明对所有$n \ge 5$，$\text{dim}_m(P_n \boxtimes P_n)=4$，回答了Hakanen和Yero的问题：将坐标旋转45°后，国王网格的切比雪夫度量变为奇偶子格上曼哈顿度量的一半，且四个边界不等式将每个潜在的3个 landmark（ landmark 译为“地标点”）的分辨集简化为两种几何情况，每种情况中我们都展示了一个明确的碰撞。第三，在国王带上该参数呈线性增长：对$n \ge 6$，$\text{dim}_m(P_3 \boxtimes P_n)=n$（小阶数情况已精确确定），其中下界依赖于三个局部分离条件和一个有限的 min-plus 转移证书，其等号情况产生一个具有19个状态的循环核心的有限自动机，而上界是一个对所有高度都有效的周期为3的明确地标模式。结合 blindness（盲性）下界，对每个固定的$h \ge 3$，$\text{dim}_m(P_h \boxtimes P_n) = \Theta(n)$，因此正方形国王网格上的常数结果需要两个维度都增长。
英文摘要
We present three results on the multiset dimension of graphs, resolving one conjecture and two open questions from the literature. First, we disprove the conjecture of Simanjuntak, Siagian and Vetrík (2017) that every graph $G$ of order $n(G)$ with finite multiset dimension satisfies $\dim_m(G) \le n(G)-1$: an exhaustive computation over all 1,018,690,328 connected graphs of orders 2 through 11 shows that exactly eight graphs attain $\dim_m(G)=n(G)$, all of order 11, so 11 is the smallest order at which the trivial upper bound is attained. This also answers a question from the recent survey of Farhan, Klavžar, Kuziak and Yero. Second, we prove that $\dim_m(P_n \boxtimes P_n)=4$ for every $n \ge 5$, answering a question of Hakanen and Yero: after a $45^\circ$ change of coordinates the Chebyshev metric of the king grid becomes half the Manhattan metric on a parity sublattice, and four boundary inequalities reduce every potentially resolving three-landmark set to two geometric cases, in each of which we exhibit an explicit collision. Third, on king strips the parameter grows linearly: $\dim_m(P_3 \boxtimes P_n)=n$ for $n \ge 6$ (with the small cases determined exactly), where the lower bound rests on three local separation conditions and a finite min-plus transfer certificate whose equality case yields a finite automaton with a 19-state recurrent core, and the upper bound is an explicit landmark pattern of period three that works for every height. Combined with a blindness lower bound, $\dim_m(P_h \boxtimes P_n) = Θ(n)$ for every fixed $h \ge 3$, so the constant answer on square king grids requires both dimensions to grow.
Comments22 pages. Consolidates three related manuscripts into one at the request of the arXiv moderators. Code, data and certificates accompany as ancillary files and are permanently archived at doi:10.5281/zenodo.21612126 (Part I), doi:10.5281/zenodo.21576586 (Part II), doi:10.5281/zenodo.21609917 (Part III)
