On factors of 4-connected claw-free graphs (Q2744573)

This paper is motivated by the following two conjectures. Conjecture 1 (\textit{C. Thomassen} [J. Graph Theory 10, 309-324 (1986; Zbl 0614.05050)]): Every 4-connected line graph is Hamiltonian. Conjecture 2 (\textit{M. M. Matthews} and \textit{D. P. Sumner} [J. Graph Theory 8, 139-146 (1984; Zbl 0536.05047)]): Every 4-connected claw-free graph is Hamiltonian. Since line graphs are claw-free, Conjecture 2 implies Conjecture 1. In 1997, the third author of the present paper showed that these conjectures are equivalent. In the present paper the authors show that Conjecture 2 is true within the class of so-called hourglass-free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. In addition, the authors prove the following weaker form of Conjecture 2. Every 4-connected claw-free graph contains a connected factor which has degree 2 or 4 at each vertex.