Closure and stable Hamiltonian properties in claw-free graphs

In the class of k-connected claw-free graphs, we study the stability of some hamilto-nian 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 innnite 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 paper is concluded with several open questions concerning stability of homege-neous traceability and hamiltonian connectedness.