Disjoint Chorded Cycles of the Same Length

Bollobas 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 Verstraete's result: A graph of sufficiently large order with minimum degree at least $2k$ contains $k$ vertex-disjoint cycles of the same length.