AI 中文总结
该研究构造了每个开邻域同构于K(7,3)的240顶点连通图，通过上同调分类固定基的二元电压覆盖，经S₁₀作用商后得到1245395个轨道，明确了目标图的上同调类及稳定子群结构。
AI 中文摘要
我们构造了一个含240个顶点的连通图，其每个开邻域均同构于K(7,3)。该图源于K(10,3)的二元电压覆盖。更一般地，在固定带标记基K(10,3)上，保持局部邻域的二元电压覆盖的规范类，自然对应于H¹(M₃(10);𝔽₂)，这是一个维数为42的向量空间。对自然的S₁₀作用取商后得到1245395个轨道，其中包括1245394个非零轨道，每个非零轨道都由连通覆盖组成。所示240顶点图的上同调类[α]具有S₁₀-轨道大小126，且Stab_{S₁₀}([α])≅S₅≀C₂。因此，固定基覆盖可通过上同调分类，而允许基重标记则得到上述S₁₀-轨道集合。
英文摘要
For every $d\geq3$ we construct a connected graph of order $2\binom{3d+1}{d}$ that is locally $K(2d+1,d)$. Fix a block $A$ and put $a'=\min(|A|,3d+1-|A|)$. Among the loopless binary voltages on the fixed labeled base $K(3d+1,d)$ that depend only on the intersection types with $A$, the local-neighborhood-preserving assignments form, modulo gauge, an $\mathbb F_2$-space of dimension $[q^{a'-5}]\binom{d}{3}_q$, with explicit canonical coordinates. Adjacent-line rigidity holds on $2d+2\leq n\leq3d$: every loopless fixed-block type-invariant local-neighborhood-preserving binary voltage is gauge trivial. Dropping type invariance over the fixed labeled $K(10,3)$, we classify all local-neighborhood-preserving binary voltage assignments and find that $H^1(M_3(10);\mathbb F_2)$ has dimension 42. Under base relabeling by $S_{10}$ these classes form 1,245,395 orbits, of which 1,245,394 consist of connected covers; an explicit class has orbit size 126 and stabilizer $S_5\wr C_2$. The proofs combine a triangle-voltage criterion and cohomology with a simplicial collapse and Burnside enumeration.
Comments21 pages. Substantially expanded from v1. Adds a fixed-block type-voltage construction, an infinite family for n=2d+1, and rigidity for 2d+2 <= n <= 3d within this framework. Retains the K(10,3) cohomology and symmetry-orbit classification. Reproducibility materials: https://doi.org/10.5281/zenodo.22092735
