Type-Voltage Covers and Finite Locally Kneser Graphs

Abstract

For every integer d ≥ 3, we construct a connected graph on twice the binomial coefficient ‘3d + 1 choose d’ vertices that is locally the Kneser graph K(2d + 1, d). The construction on the line n = 2d + 1 uses a binary voltage cover of K(3d + 1, d), with voltage determined only by intersection types with a fixed subset A. Up to gauge equivalence, the loopless type-invariant binary voltage assignments form a quotient space whose dimension is the coefficient of q to the power a′ − 5 in the Gaussian binomial coefficient ‘d choose 3’, where a′ is the smaller of |A| and 3d + 1 − |A|. Every nonzero class gives a connected cover, and taking |A| = 5 and assigning voltage one exactly to base edges joining types 1 and 2 gives the uniform family. By contrast, throughout the adjacent rigidity interval 2d + 2 ≤ n ≤ 3d, every loopless type-invariant binary voltage on K(n + d, d) that preserves local neighborhoods is gauge equivalent to zero. The rigidity conclusion follows from an explicit simplicial collapse and induction.