A generalization of Dirac's theorem on cycles through k vertices in k-connected graphs

Let X be a subset of the vertex set of a graph G. We denote by @k(X) the smallest number of vertices separating two vertices of X if X does not induce a complete subgraph of G, otherwise we put @k(X)=|X|-1 if |X|>=2 and @k(X)=1 if |X|=1. We prove that if @k(X)>=2 then every set of at most @k(X) vertices of X is contained in a cycle of G. Thus, we generalize a similar result of Dirac. Applying this theorem we improve our previous result involving an Ore-type condition and give another proof of a slightly improved version of a theorem of Broersma et al.