Hypergeometric closed forms for unicellular Ising cubic maps with a fixed number of bicolored 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 R_{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,3n−i,j} = R_{i,j}(g) · (6g)!/(12^{g}·g!·(3g)!) holds; additionally, show vanishing when i+j is not a multiple of 3 or when j>i. This would provide hypergeometric-type closed forms for all cases with a fixed number of bicolored edges (equal to i−j).

Background

Let U be the Ising generating function of half-edge-labeled unicellular (one-face) cubic maps, where variables mark the number of white- and black-monochromatic edges. The coefficients U_{n,k,ℓ} count such maps with 3n edges, k white-monochromatic edges, and ℓ black-monochromatic edges.

The paper derives explicit hypergeometric expressions in several cases (e.g., all edges monochromatic, or one bicolored edge) and provides further identities for small values of i+j, suggesting a broader pattern. The stated conjecture formulates a general hypergeometric closed form depending on a rational function R_{i,j}(g) for all fixed i and j, with natural vanishing constraints based on parity and color-balance conditions.