Séminaire Lotharingien de Combinatoire, B30i (1993), 6 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1993, 1993/034, p.
29-34.]
Dietrich Gernert
Partial Proofs of Thomassen's Conjecture
Abstract.
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.
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