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Hamiltonian connectedness in 3-connected line graphs. (English) Zbl 1169.05344
Summary: 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. Comb. Theory, Ser. B 82, No. 2, 306–315 (2001; Zbl 1027.05059)] 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 a 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 $$\kappa (L(G))\geq 3$$, which extends a recent result by Kriesell [cited above] that every 4-connected hourglass free line graph is hamiltonian connected.

##### MSC:
 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs
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