Hamiltonian circles and Hamiltonian curves

    A closed curve in the Freudenthal compactification |G| of an infinite locally finite graph G is called a Hamiltonian curve if it meets every vertex of G exactly once (and hence it meets every end at least once). If moreover it meets every end of G exactly once, then it is a Hamiltonian circle. We prove that |G| has a Hamiltonian curve if and only if every Finite vertex set of G is contained in a cycle of G. We apply this to extend a number of results and conjectures on finite graphs to Hamiltonian curves in infinite locally finite graphs. We also give some conditions for the existence of Hamiltonian circles of infinite locally finite graphs.  (joint work with André Kündgen and Carsten Thomassen)