Geodesic
Partial Proofs of Thomassen's Conjecture
Séminaire lotharingien de combinatoire, Tome 30 (1993) Cet article a éte moissonné depuis la source Séminaire Lotharingien de Combinatoire website

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THOMASSEN's Conjecture (1986) claims that every line-graph with vertex-connectivity number κ ≥ 4 is hamiltonian. This conjecture is equivalent to several other conjectures. The partial proofs which will be summed up here do not start directly from a line-graph, but from a graph «I>G such that its line-graph L(G) has the properties as above. It will be shown that both G and L(G) must fulfill a great number of restrictions if L(G) is to be a counterexample to THOMASSEN's Conjecture. These restrictions are both structural properties and inequalities related to several graph invariants. It is proved e.g. that a counterexample must have at least 23 vertices, it has a 2-cover, and it is not locally connected. 

@article{SLC_1993_30_a8,
     author = {Dietrich Gernert},
     title = {Partial {Proofs} of {Thomassen's} {Conjecture}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {1993},
     volume = {30},
     url = {http://geodesic.mathdoc.fr/item/SLC_1993_30_a8/}
}
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Dietrich Gernert. Partial Proofs of Thomassen's Conjecture. Séminaire lotharingien de combinatoire, Tome 30 (1993). http://geodesic.mathdoc.fr/item/SLC_1993_30_a8/