The Genus of Bipartite Kneser Graphs

We determine the orientable genus of an infinite family of bipartite Kneser graphs. The graph $H(h,2)$ has two copies of the two-element subsets of $[h]$, with opposite-class vertices adjacent when the corresponding subsets are disjoint. For every prime $h>3$ with $h\equiv3\pmod8$, we prove $$\gamma(H(h,2))=1-\frac{h(h-1)}{2}+\frac{h(h-1)(h-2)(h-3)}{16}.$$ Euler's formula gives this lower bound, with equality for a quadrangulation. We construct a vertex-transitive orientable quadrangulation using an odd-order affine group that acts simply transitively on the two-element subsets. This gives an infinite family satisfying Pisanski's conjecture on quadrilateral embeddings of regular bipartite graphs.

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