A remark on embedded bipartite graphs (Q1072561)

Let \(m\) be the total number of \(4k\)-gonal faces of a bipartite graph embedded on an orientable surface. \textit{J. Zaks} [J. Comb. Theory, Ser. B 32, 95--98 (1982; Zbl 0485.05036)] used the Euler formula to conclude that, if the surface is the sphere, \(m\equiv v\pmod 2\), where \(v\) is the number of vertices of the graph. This note provides a proof of this relation -- for an arbitrary orientable surface -- which replaces the use of the Euler formula by parity consideration of three permutations of the set of edges of the graph, naturally associated with the embedding, whose product is the identity.