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Let G be a graph. For u,v∈V(G) with distG(u,v)=2, denote JG(u,v)={w∈NG(u)∩NG(v)|NG(w)⊆NG(u)∪NG(v)∪{u,v}}. A graph G is called quasi claw-free if JG(u,v)≠∅ for any u,v∈V(G) with distG(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected.
Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309–324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5-E(C4), where C4 is a cycle of length 4 in K5. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125–136], it is shown that every 4-connected line graph without an induced subgraph isomorphic...
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