Euler-bound conjecture for bipartite Kneser graphs

Prove that for every integer n≥7, the orientable genus of the bipartite Kneser graph H(n,2) equals the rounded Euler lower bound γ(H(n,2))=⌈1+(1/4)\binom{n}{2}(\binom{n-2}{2}-4)⌉.

Background

The paper establishes the exact orientable genus of H(h,2) for primes h>3 satisfying h≡3 (mod 8), by constructing vertex-transitive orientable quadrangulations that attain the Euler lower bound. Computations also show equality with the rounded Euler bound for every 7≤n≤20, while the known k=1 formula has the analogous property for all n≥3.

The conjecture extends these results to every n≥7. For n≡0,1,2,3 (mod 8), the lower bound is integral and the conjecture requires an orientable quadrangulation of H(n,2). For n≡4,5,6,7 (mod 8), it requires an orientable embedding with all faces quadrilateral except for either two hexagons or one octagon.