Sharp minimum degree conditions for disjoint doubly chorded cycles

In 1963, Corradi and Hajnal proved that if G is an n -vertex graph where n >= 3 k and delta ( G ) >= 2 k , then G contains k vertex -disjoint cycles, and furthermore, the minimum degree condition is best possible for all n and k where n >= 3 k . This serves as the motivation behind many results regarding best possible conditions that guarantee the existence of a fixed number of disjoint structures in graphs. For doubly chorded cycles, Qiao and Zhang proved that if n >= 4 k and delta ( G ) >= [ 7k /2 ], then G contains k vertex -disjoint doubly chorded cycles. However, the minimum degree in this result is sharp for only a finite number of values of k . Later, Gould Hirohata, and Horn improved upon this by showing that if n >= 6 k and delta ( G ) > 3 k , then G contains k vertex -disjoint doubly chorded cycles. Furthermore, this minimum degree condition is best possible for all n and k where n >= 6 k . In this paper, we prove two results. First, we extend the result of Gould et al. by showing their minimum degree condition guarantees k disjoint doubly chorded cycles even when n >= 5 k , and in addition, this is best possible for all n and k where n >= 5 k . Second, we improve upon the result of Qiao and Zhang by showing that every n -vertex graph G with n >= 4 k and delta ( G ) >= [10 k - 1/3] , contains k vertex -disjoint doubly chorded cycles. Moreover, this minimum degree is best possible for all k E Z (+) .