We consider the existence of several different kinds of factors in 4-connected211u001eclaw-free graphs. This is motivated by the following conjectures which are in 211u001efact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): 211u001eEvery 4-connected line graph is hamiltonian, i.e. has a connected 2-factor. 211u001eConjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is 211u001ehamiltonian. We first show that Conjecture 2 is true within the class of 211u001ehourglass-free graphs, i.e. graphs that do not contain an induced subgraph 211u001eisomorphic to two triangles meeting in exactly one vertex. Next we show that a 211u001eweaker form of Conjecture 2 is true, in which the conclusion is replaced by the 211u001econclusion that there exists a connected spanning subgraph in which each vertex 211u001ehas degree two or four. Finally we show that Conjecture 1 and 2 are equivalent to 211u001eseemingly weaker conjectures in which the conclusion is replaced by the 211u001econclusion that there exists a spanning subgraph consisting of a bounded number 211u001eof paths.
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