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April 3, 2026Open Access

The exact second Betti number of the neighborhood complex of the Kneser graph KG(2k+2,k)

Key Points

To determine and verify the second Betti number of the neighborhood complex of the Kneser graph KG(2k+2,k).
Computed the Euler characteristic through combinatorial face characterization.
Applied Björner's theorem on homotopy type for verification.
Extended conjectured results computationally for values of k from 2 to 7.
The formula for beta_2 is given by: C(2k+1,k)(k^3-k+2)/(k+2) - 1 for all k ≥ 2.
Validated previously conjectured results for k=2,...,7 with computational support.
Established new results for k ≥ 4, building on earlier work by Nilakantan-Singh.

Abstract

We prove that the second Betti number of the neighborhood complex of the Kneser graph KG (2k+2, k) is given by the exact formula beta₂ = C (2k+1, k) (k³-k+2) / (k+2) - 1 for all k >= 2. The proof proceeds via an explicit computation of the Euler characteristic using a combinatorial characterization of faces, together with Björner's theorem on the homotopy type. Previously conjectured and verified computationally for k=2,. . . , 7 (including a 6. 8-hour C-accelerated computation for KG (16, 7) yielding beta₂=241, 669), this formula is now a theorem. The result is new for k >= 4, extending the work of Nilakantan-Singh (2018) who covered k=2, 3 by different methods.

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