Disjoint chorded cycles in graphs

AbstractWe propose the following conjecture to generalize results of Pósa and of Corrádi and Hajnal. Let r,s be nonnegative integers and let G be a graph with |V(G)|≥3r+4s and minimal degree δ(G)≥2r+3s. Then G contains a collection of r+s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r=0,s=2 and for s=1. The corresponding extremal problem, to find the minimum number of edges in a graph on n vertices ensuring the existence of two vertex disjoint chorded cycles, is also settled