Closure and stable Hamiltonian properties in claw-free graphs

In the class of k-connected claw-free graphs, we study the stability of some Hamiltonian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e,, for any of these properties there is an infinite family of graphs G(k) of arbitrarily high connectivity k such that the closure of G(k) has the property while the graph G(k) does not); (ii) traceability is a stable property even for k = 1; (iii) homogeneous traceability is not stable for k = 2 (although it is stable for k = 7). The article is concluded with several open questions concerning stability of homogeneous traceability and Hamiltonian connectedness. (C) 2000 John Wiley & Sons, Inc.