We consider the existence of several different kinds of factors in 4-connec
ted claw-free graphs. This is motivated by the following two conjectures wh
ich are in fact equivalent by a recent result of the third author. Conjectu
re 1 (Thomassen): Every 4-connected line graph is hamiltonian, i.e., has a
connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected c
law-free graph is hamiltonian. We first show that Conjecture 2 is true with
in the class of hourglass-free graphs, i.e., graphs that do not contain an
induced subgraph isomorphic to two triangles meeting in exactly one vertex.
Next we show that a weaker form of Conjecture 2 is true, in which the conc
lusion is replaced by the conclusion that there exists a connected spanning
subgraph in which each vertex has degree two or four. Finally we show that
Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in whic
h the conclusion is replaced by the conclusion that there exists a spanning
subgraph consisting of a bounded number of paths. (C) 2001 John Wiley & So
ns, Inc. J Graph Theory 37: 125-136, 2001.