Type-Voltage Covers and Finite Locally Kneser Graphs

Abstract

For every $d\ge 3$ we construct a connected graph of order $2\left(\genfrac{}{}{0ex}{}{3d+1}{d}\right)$ that is locally $K(2d+1,d)$. Fix a block $A$ and put $a^{\prime }=\min (|A|,3d+1-|A|)$. 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 $[q^{a^{\prime }-5}]{\left(\genfrac{}{}{0ex}{}{d}{3}\right)}_q$, with explicit canonical coordinates. Adjacent-line rigidity holds on $2d+2\le n\le 3d$: 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.

21 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