Type-Voltage Covers and Finite Locally Kneser Graphs

Abstract

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.