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 (X) the smallest number of vertices separating two vertices of X if X does not induce a complete subgraph of G, otherwise we put (X) = |X| − 1 if |X| 2 and (X) = 1 if |X| = 1. We prove that if (X) 2 then every set of at most (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. © 2006 Elsevier B.V. All rights reserved.