Combinatorics of loop equations for branched covers of sphere
  We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion. 
P. Dunin-Barkowski, N. Orantin, A. Popolitov, S. Shadrin
math-ph math.CO math.MP