Two-Face Complete Regular Dessins

A complete dessin is an orientable map with underlying graph being a complete bipartite graph, which is said to be regular if its group of color- and orientation-preserving automorphisms acts regularly on the edges. This paper presents a classification and enumeration of complete regular dessins that have exactly two faces, which is not only a generalization of uniface regular dessins (see Fan et al. in J Algebr Comb 49:125–134, 2019) and planar two-face maps (see Bousquet et al. in Discrete Math 222:1–25, 2000), but also a special case of the classification of complete regular dessins proposed by Jones.