Cycle‐minors and subdivisions of wheels

In this paper we prove two results. The first is an extension of a result of Dirac which says that any set of n vertices of an n ‐connected graph lies in a cycle. We prove that if V ′ is a set of at most 2 n vertices in an n ‐connected graph G , then G has, as a minor, a cycle using all of the vertices of V ′. The second result says that if G is an n +1‐connected graph with maximum vertex degree Δ then G contains a subgraph that is a subdivision of W 2 n if and only if Δ≥2 n . © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 100–108, 2009