On the Thomassen's conjecture

C. Thomassen proposed a conjecture: Let G be a k-connected graph with the s tability number alpha greater than or equal to k, then G has a cycle C cont aining k independent vertices and all their neighbors. In this paper, we wi ll obtain the following result: Let G be a k-connected graph with stability number alpha = k + 3 and C any longest cycle of G, then C contains k indep endent vertices and all their neighbors. This solves Thomassen's conjecture for the case alpha = k + 3. (C) 2001 John Wiley & Sons, Inc.