Extending cycles locally to Hamilton cycles (Q278887)

Summary: A Hamilton circle in an infinite graph is a homeomorphic copy of the~ unit circle \(S^1\) that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of \textit{D. J. Oberly} and \textit{D. P. Sumner} [J. Graph Theory 3, 351--356 (1979; Zbl 0424.05036)] to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are \(3\)-connected, extending a result of \textit{A. S. Asratian} [J. Graph Theory 23, No. 2, 191--201 (1996; Zbl 0858.05067)]. \textit{Hamilton-connected} means that between any two vertices there is a Hamilton arc, a homeomorphic copy of the unit interval \([0,1]\) that contains all vertices and all ends precisely once.