Book Embedding of Toroidal Bipartite Graphs

Endo [5] proved that every toroidal graph has a book embedding with at most seven pages. In this paper, we prove that every toroidal bipartite graph has a book embedding with at most five pages. In order to do so, we prove that every bipartite torus quadrangulation Q with n vertices admits two disjoint essential simple closed curves cutting the torus into two annuli so that each of the two annuli contains a spanning connected subgraph of Q with exactly n edges.