# Infinite uniquely hamiltonian graphs

**Problem**Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree ?

(Originally appeared as [M].)

Let be a locally finite infinite graph and let be the set of ends of~. The Freudenthal compactification of is the topological space which is obtained from the usual topological space of the graph, when viewed as a 1-dimensional cell complex, by adding all points of and setting, for each end , the basic set of neighborhoods of to consist of sets of the form , where ranges over the finite subsets of , is the component of containing all rays in , the set contains all ends in having rays in , and is the union of half-edges , one for every edge joining and . We define a *hamilton circle* in as a homeomorphic image of the unit circle into such that every vertex (and hence every end) of appears in . More details about these notions can be found in [D].

A graph (finite or infinite) is said to be uniquely hamiltonian if it contains precisely one hamilton circle.

For finite graphs, the celebrated Sheehan's conjecture states that there are no -regular uniquely hamiltonian graphs for ; this is known for all odd and even . For infinite graphs this is false even for odd (e.g. for the two-way infinite ladder), but each of the known counterexamples has at least 2 ends, leading to the problem stated.

Another way to extend Sheehan's conjecture to infinite graphs is to define *degree of an end * to be the maximal number of disjoint rays in and ask the following:

**Problem**Are there any uniquely hamiltonian locally finite graphs where every vertex and every end has the same degree ?

## Bibliography

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

*[M] Bojan Mohar, Problem of the Month

* indicates original appearance(s) of problem.