# Which graphs satisfy this property?

I am currently looking into a conjecture in graph theory, known as the Jackson conjecture (1992). It says the following, in an equivalent formulation to its equivalent statement:

*Conjecture*Every 4-connected claw-free graph is Hamiltonian.

Recall that a graph is hamiltonian if a cycle visits each vertex exactly once, and that it is $n$-connected given that the deletion of those $n$ vertices keeps the graph together.

My question is, is there any reference that summarizes all of the 4-connected claw-free graphs?, i.e., is there any paper out there with a list of those graphs?

Thanks.

-
 This is actually known as the Matthews-Sumner conjecture. The paper in which it first appeared, entitled "Longest paths and cycles in $K_{1,3}$-free graphs", is available here. – Andrew Uzzell Dec 11 '12 at 13:14