Let G be a locally finite infinite graph and let I(G) be the set of ends of G. The Freudenthal compactification of G is the topological space |G| which is obtained from the usual topological space of the graph, when viewed as a 1-dimensional cell complex, by adding all points of I(G) and setting, for each end in I(G), the basic set of neighborhoods of t to consist of sets of the form C(S, t) I(S,t) E'(S,t), where S ranges over the finite subsets of V(G), C(S, t) is the component of G - S containing all rays in t, the set I(S,t) contains all ends in I(G) having rays in C(S, t), and E'(S,t) is the union of half-edges (z,y], one for every edge xy joining S and C(S,t). We define a hamilton circle in |G| as a homeomorphic image C of the unit circle S1 into |G| such that every vertex (and hence every end) of G appears in C. More details about these notions can be found in .
A graph G (finite or infinite) is said to be uniquely hamiltonian if it contains precisely one hamilton circle.
For finite graphs, Sheehan proposed the following conjecture in 1975:
Conjecture 1 (Sheehan ): If G is a finite r-regular graph, where r > 2, then G is not uniquely hamiltonian.
This conjecture has been proved for all odd values of r  and for all even values of r > 23 . By Petersen's theorem, it would suffice to prove it for r = 4.
For infinite graphs, Conjecture 1 is false even for odd values of r, however all known counterexamples (e.g. the two-way infinite ladder) have at least 2 ends. Thus, one can ask if it could be true for 1-ended graphs:
Problem 2: Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree r > 2?
Another possiblility is to ask if conjecture 1 could be true even for graphs with many ends if we assume that (in addition to the vertices) the ends have a certain vertex-degree (the vertex-degree of an end t is the maximal number of disjoint rays in t).
Several other problems about hamilton circles in infinite graphs have been proposed by Georgakopoulos:
Conjecture 3 (Georgakopoulos): If G is a 4-edge-connected locally finite graph, then its line graph is hamiltonian.
This 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 ) 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).
Conjecture 4 (Georgakopoulos): If the line graph L(G) of a locally finite graph G is 4-connected, then L(G) is hamiltonian.
Conjecture 4 is widely open even in the finite case, where it was proposed by Thomassen .
For finite graphs it is well known that the third power of a connected graph is hamiltonian. This has also been proved for locally finite graphs . But what about non-locally-finite graphs?
Conjecture 5 (Georgakopoulos ): If G is a countable connected graph then its third power is hamiltonian.
Similarly, one can ask if Fleischner's theorem holds for countable graphs (for locally finite ones it has already been proved ):
Conjecture 6 (Georgakopoulos ): If G is a 2-connected countable graph then its square is hamiltonian.
 R. Diestel, Graph Theory, Third Edition, Springer, 2005.
 P. Haxell, B. Seamone, J. Verstraete, Independent dominating sets and hamiltonian cycles, J. Graph Theory 54 (2007) 233-244.
 J. Sheehan, The multiplicity of Hamiltonian circuits in a graph. Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pp. 477-480. Academia, Prague, 1975.
 A.G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). Ann. Discrete Math. 3 (1978), Exp. No. 13, 3 pp.
 C. Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324.
 A. Georgakopoulos, Oberwolfach reports, 16/2007.
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