A generalization of Diracʹs theorem on cycles through k vertices in k-connected graphs Original Research Article

Let X be a subset of the vertex set of a graph G. We denote by image the smallest number of vertices separating two vertices of X if X does not induce a complete subgraph of G, otherwise we put image if image and image if image. We prove that if image then every set of at most image 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.