AI 中文总结
该研究解决超立方体边多重集维数的有限性问题，确定其无穷性对应d=2至5，给出Q_6至Q_10的分辨集，对d≥11用概率方法证明存在性并覆盖d=11至50及d≥51，还证得Q_6的下界。
AI 中文摘要
对于图G和非空顶点集S，边e的边多重集表示是e到S中各元素距离的多重集，其中d(uv,s)=min{d(u,s),d(v,s)}。边多重集维数edim_m(G)是使得所有边表示两两不同的集合的最小基数，若不存在这样的集合则其为无穷大。近期一项综述提出问题：对所有d≥3，edim_m(Q_d)是否为无穷大？我们否定了该问题并完全确定了有限-无穷的转变：edim_m(Q_d)为无穷大当且仅当2≤d≤5。穷尽计算证明edim_m(Q_5)=∞，扩展了已知的Q_3和Q_4的不存在性结果；针对Q_6至Q_10，给出了显式且可独立验证的分辨集；对于所有d≥11，我们概率性地证明其存在性：两条随机边直方图相等是距离层图上的零散度事件，对生成森林外的条件将其概率用中心二项式原子的乘积界定；经认证的精确有理计算覆盖11≤d≤50，而初等十边森林估计处理尾部d≥51。我们还证明了下界edim_m(Q_6)≥6。
英文摘要
For a graph G and a nonempty set S of vertices, the edge multiset representation of an edge e is the multiset of distances from e to the elements of S, where d(uv,s)=min{d(u,s),d(v,s)}. The edge multiset dimension edim_m(G) is the minimum cardinality of a set whose edge representations are pairwise distinct, and is infinite if no such set exists. A recent survey asked whether edim_m(Q_d) is infinite for every d >= 3. We answer this question negatively and determine the finite-infinite transition completely: edim_m(Q_d) is infinite if and only if 2 <= d <= 5. An exhaustive computation proves edim_m(Q_5) = infinity, extending the known nonexistence results for Q_3 and Q_4. Explicit independently verifiable resolving sets are given for Q_6 through Q_10. For all d >= 11 we prove existence probabilistically: equality of two random edge histograms is a zero-divergence event on a graph of distance levels, and conditioning outside a spanning forest bounds its probability by a product of central-binomial atoms. Certified exact rational computations cover 11 <= d <= 50, and an elementary ten-edge forest estimate handles the tail d >= 51. We also prove the lower bound edim_m(Q_6) >= 6.
Comments13 pages. All code, certificates, orbit representatives, collision witnesses, and exact rational bounds are archived at doi:10.5281/zenodo.21739363
