# Hamiltonian cycles in line graphs of infinite graphs

\begin{conjecture} \begin{enumerate} \item If $G$ is a 4-edge-connected locally finite graph, then its \Def{line graph} is hamiltonian. \item If the line graph $L(G)$ of a locally finite graph $G$ is 4-connected, then $L(G)$ is hamiltonian. \end{enumerate} \end{conjecture}

(Reproduced from [M].)

A locally finite graph is hamiltonian, if its Freudenthal compactification (also called the end compactification, see [D]) contains a hamilton circle, i.e. a homeomorphic copy of $S^1$ containing all vertices.

The first part is known for finite graphs. The proof uses the existence of two edge-disjoint spanning trees in 4-edge-connected graphs. In the infinite case, it would be enough to prove that a 4-edge-connected locally finite graph $G$ has two edge-disjoint topological spanning trees (see [D]), one of which is connected as a subgraph of $G$. The problem is open even for the 1-ended case (where hamilton circles correspond to 2-way-infinite paths).

The second part is widely open even in the finite case, where it was proposed by \OPrefnum[Thomassen]{485} [T].

## Bibliography

[D] Reinhard Diestel, Graph Theory, Third Edition, Springer, 2005.

*[G] A. Georgakopoulos, Oberwolfach reports, 2007.

[M] Bojan Mohar, \href[Problem of the Month]{http://www.fmf.uni-lj.si/~mohar/Problems/P0703_HamiltonicityInfinite.html}

[T] Carsten Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324, \MRhref{MR0856118}

* indicates original appearance(s) of problem.