Hypergeometric closed forms for unicellular Ising cubic maps with few monochromatic edges

Establish, for unicellular cubic maps endowed with the Ising model, that for every fixed pair of nonnegative integers i and j, there exists a rational function Q_{i,j}(g) in the genus g such that, with n=2g−1 and U_{n,k,ℓ} denoting the coefficient of t^{3n} marking k white-monochromatic edges and ℓ black-monochromatic edges in the unicellular Ising generating function U(t,·,·), the identity U_{n,i,j} = Q_{i,j}(g) · (2g)!·(3g)!/(3^{g}·g!^{3}) holds; additionally, show vanishing when i−j is not a multiple of 3. This would yield hypergeometric-type closed forms when the number of monochromatic edges is small.

Background

Beyond the regime with a fixed number of bicolored edges, the authors present evidence and explicit identities suggesting hypergeometric formulas when the total number of monochromatic edges is small. They propose a general form involving a rational function Q_{i,j}(g) multiplying a hypergeometric term, with natural congruence constraints for nonvanishing.

Several computed cases (e.g., bipartite extremal counts and configurations with a handful of monochromatic edges) support this pattern, but a general proof is not known.