Hamiltonian connectedness in 3-connected line graphs

Abstract

We investigate graphs G such that the line graph L ( G ) is hamiltonian connected if and only if L ( G ) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G , then L ( G ) 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 L ( G ) does not have an hourglass (a graph isomorphic to K 5 − E ( C 4 ) , where C 4 is an cycle of length 4 in K 5 ) as an induced subgraph, and if every 3-cut of L ( G ) is not independent, then L ( G ) is hamiltonian connected if and only if κ ( L ( G ) ) ≥ 3 , 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.