AbstractBondy conjectured that if G is a k‐connected graph of order n such that magnified image for any (k + 1)‐independent set / of G, then the subgraph outside any longest cycle contains no path of length k − 1. In this paper, we are going to prove that, if G is a k‐connected claw‐free (K1,3‐free) graph of order n such that magnified image for any (k + 1)‐independent set /, then G contains a Hamilton cycle. The theorem in this paper implies Bondy's conjecture in the case of claw‐free graphs.