Hamiltonian connectedness in 3-connected line graphs.

*(English)*Zbl 1169.05344Summary: 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.

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\textit{H.-J. Lai} et al., Discrete Appl. Math. 157, No. 5, 982--990 (2009; Zbl 1169.05344)

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