Disjoint chorded cycles in graphs

We propose the following conjecture to generalize results of Pósa and of Corrádi and Hajnal. Let image be nonnegative integers and let image be a graph with image and minimal degree image. Then image contains a collection of image vertex disjoint cycles, image of them with a chord. We prove the conjecture for image and for image. The corresponding extremal problem, to find the minimum number of edges in a graph on image vertices ensuring the existence of two vertex disjoint chorded cycles, is also settled.