Enumeration of 4-regular one-face maps

We obtain explicit formulas for enumerating rooted and unrooted 4-regular one-face maps on genus g surfaces. For rooted maps the result is combinatorially derived from Chapuy's vertex cutting bijection and has a simple sum-free form similar to analogous formulas for general and cubic one-face maps. To enumerate unrooted maps we apply the approach of Liskovets, Mednykh and Nedela of reducing the problem to counting rooted maps on orbifolds. We show that for 4-regular one-face maps the set of orbifolds to be considered has a simple description, which allows us to obtain the final formula in an explicit form of a sum over 3 integer indices.