On the Thomassen's conjecture *

C. Thomassen proposed a conjecture: Let G be a k ‐connected graph with the stability number α ≥  k , then G has a cycle C containing k independent vertices and all their neighbors. In this paper, we will obtain the following result: Let G be a k ‐connected graph with stability number α =  k  + 3 and C any longest cycle of G , then C contains k independent vertices and all their neighbors. This solves Thomassen's conjecture for the case α =  k  + 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 168–180, 2001