# Hamiltonian cycles in line graphs

 Importance: High ✭✭✭
 Author(s): Thomassen, Carsten
 Subject: Graph Theory » Basic Graph Theory » » Cycles
 Keywords: hamiltonian line graphs
 Posted by: Robert Samal on: July 24th, 2007
Conjecture   Every 4-connected line graph is hamiltonian.
\item It is known that if is 4-edge-connected, then its line graph is hamiltonian. \item Thomassen's is a special case of a conjecture due to Matthews and Sumner: every 4-connected claw-free graph is hamiltonian. \item However, by a result of Ryjacek [R] conjectures of Thomassen and of Matthews and Sumner are equivalent. \item Moreover [R], one may restrict to 4-connected line graphs of triangle-free graphs.

## Bibliography

[R] Zdenek Ryjacek: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70 (1997), no. 2, 217--224, MathSciNet

*[T] Carsten Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324, MathSciNet

* indicates original appearance(s) of problem.