Establishing or refuting Thomassen's conjecture remains an difficult open problem, as attested to by the fact that the middle
levels conjecture, which posited that middle
level graphs are HamitlonianHamiltonian Graph,
was proven only relatively recently ((Mütze 2016, Mütze 2024).

A slightly weaker conjecture is that all Cayley graphs are Hamiltonian (Royle). Conversely, all Cayley graphs
are vertex-transitive.

Alspach (1979) showed that every connected vertex-transitive graph of order except the Petersen graph
is Hamiltonian. Marušič (1982) showed
that every connected vertex-transitive graph of order , , , and is Hamiltonian, while
Marušič and Parsons (1983) showed that connected vertex-transitive graphs
of order and are traceable (Gould 1991).

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Graphs of Order ." Disc. Math.42, 227-242, 1982.Marušič,
D. and Parsons, T. D. "Hamiltonian Paths in Vertex-Symmetric Graphs of
Order ." Disc. Math.43, 91-96, 1983.Mütze,
T. "Proof of the Middle Levels Conjecture." Proc. Lond. Math. Soc.112,
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by Intersecting Set Systems." Not. Amer. Soc.74, 583-592, 2024.Royle,
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