Vertex-transitive quadrangulations beyond the prime-power cases

Characterize, for integers n≥8 with n≡0,1,2,3 (mod 8), exactly when the bipartite Kneser graph H(n,2) admits a vertex-transitive quadrangulation.

Background

The paper constructs vertex-transitive quadrangulations for primes h≡3 (mod 8). It explains that when n≡4,5,6,7 (mod 8), an embedding attaining the rounded Euler bound cannot be vertex-transitive once the graph has more than twelve vertices, because the nonquadrilateral faces would have to be preserved by automorphisms.

For n≥8 with n≡0,1,2,3 (mod 8), the existence of a vertex-transitive quadrangulation remains unresolved in general. Godsil's classification shows that a subgroup of Sym([n]) acting simply transitively on the two-element subsets exists only when n is a prime power congruent to 3 modulo 4, so other cases require different symmetry mechanisms.