An analog of the Harer-Zagier formula for unicellular bicolored maps (Q1128363)

A map is a 2-cell imbedding of a graph into a closed orientable 2-manifold; it is unicellular if there is just one 2-cell, and bicolored if the graph is bipartite. The author uses an idea of \textit{D. Zagier} [Nieuw Arch. Wiskd., IV. Ser. 13, No. 3, 489-495 (1995; Zbl 0854.05008)] to enumerate bicolored unicellular maps with prescribed numbers of edges and vertices of both colors. A recurrence relation and generating function are found, the latter being expressed in terms of the Gauss hypergeometric function. It is then shown that the fraction of bicolored unicellular maps of genus \(g\) is approximately \((1/4)^g\), if the number of edges is large enough.