Bollobás and Thomason showed that a multigraph of order $n$ and size at least $n+c\,(c\ge 1)$ contains a cycle of length at most $2(\lfloor n/c\rfloor+1)\lfloor \log_2 2c\rfloor$. We show in this paper that a multigraph (with no loop) of order $n$ and minimum degree at least 5 contains a chorded cycle (a cycle with a chord) of length at most $300\log_2 n$. As an application of this result, we show that a graph of sufficiently large order with minimum degree at least $3k+8$ contains $k$ vertex-disjoint chorded cycles of the same length, which is analogous to Verstraëte's result: A graph of sufficiently large order with minimum degree at least $2k$ contains $k$ vertex-disjoint cycles of the same length.

Endo 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 noncontractible 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 satisfying a certain condition.