Hamiltonian connectedness in 3-connected line graphs Original Research Article

We investigate graphs image such that the line graph image is hamiltonian connected if and only if image is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of image, then image has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306–315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if image does not have an hourglass (a graph isomorphic to image, where image is an cycle of length 4 in image) as an induced subgraph, and if every 3-cut of image is not independent, then image is hamiltonian connected if and only if image, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306–315] that every 4-connected hourglass free line graph is hamiltonian connected.