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Hamiltonian connected hourglass free line graphs. (English) Zbl 1147.05045
Summary: C. Thomassen [Reflections on graph theory, J. Graph Theory 10, 309–324 (1986; Zbl 0614.05050)] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to \(K_{5}-E(C_{4})\), where \(C_{4}\) is a cycle of length 4 in \(K_{5}\). In [H. J. Broersma, M. Kriesell, and Z. Ryjácek, On factors of 4-connected claw-free graphs, J. Graph Theory 37, No. 2, 125–136 (2001; Zbl 0984.05067)] it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.
MSC:
05C45 Eulerian and Hamiltonian graphs
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