Equivalence of Jackson's and Thomassen's conjectures

A graph G is said to be claw-free if G has no induced subgraph isomorphic to K1,3. For a cycle C in a graph G, C is called a Tutte cycle of G if C is a Hamilton cycle of G, or the order of C is at least 4 and every component of G − C has at most three neighbors on C. In [On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997), 217–224], Ryjáček proved that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is Hamiltonian) and by Thomassen (every 4-connected line graph is Hamiltonian) are equivalent. In this paper, we show the above conjectures are equivalent with the conjecture by Jackson in 1992 (every 2-connected claw-free graph has a Tutte cycle).