Open Problem Garden

(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 ?