Extending Cycles Locally to Hamilton Cycles

Matthias Hamann, Florian Lehner, Julian Pott

Abstract


A Hamilton circle in an infinite graph is a homeomorphic copy of theĀ  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 Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are $3$-connected, extending a result of Asratian. 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.


Keywords


Graph theory, Hamilton cycles, Infinite graphs

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