Disjoint Chorded Cycles of the Same Length | SIAM Journal on Discrete Mathematics | Vol. 29, No. 2 | Society for Industrial and Applied Mathematics

Bollobás and Thomason showed that a multigraph of order n and size at least n+ c (c ≥ 1) contains a cycle of length at most 2( n/c + 1) log2 2c . 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 log2 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.