Disjoint Chorded Cycles of the Same Length

Corrádi and Hajnal [1] showed that any graph of order at least 3k with minimum degree at least 2k contains k vertex-disjoint cycles. This minimum degree condition is sharp, because the complete bipartite graph K2k−1,n−2k+1 does not contain k vertex-disjoint cycles. About the existence of vertex-disjoint cycles of the same length, Thomassen [4] conjectured that the same minimum degree condition will suffice, if the graph has sufficiently many vertices. This conjecture was settled by Egawa [2] for k ≥ 3, and by Verstraëte [5] for all k ≥ 2. (Note that Verstraëte’s result is weaker than Egawa’s for k ≥ 3.) A chorded cycle is a cycle with an extra edge joining two nonconsecutive vertices of the cycle. By considering a longest path, one can easily see that any graph with minimum degree at least 3 contains a chorded cycle. Corresponding to Corrádi and Hajnal’s result, Finkel [3] showed that any graph with at least 4k vertices and minimum degree at least 3k contains k vertex-disjoint chorded cycles. This minimum degree condition is sharp, because the complete bipartite graph K3k−1,n−3k+1 does not contain k vertex-disjoint chorded cycles. In this talk, we combine this result with Thomassen’s conjecture. Our main result is the following.