## Abstract

A well-known conjecture of Alspach says that every (Formula presented.) -regular Cayley graph of a finite abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double-ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of (Formula presented.) -regular Cayley graphs with finite cuts (Formula presented.), where (Formula presented.) and (Formula presented.) differ in parity, which necessarily preclude the existence of a decomposition into spanning double-rays. We show that every 4-regular Cayley graph of an infinite abelian group all of whose finite cuts are even can be decomposed into spanning double-rays, and so characterise when such decompositions exist. We also characterise when such graphs can be decomposed either into Hamiltonian circles, a more topological generalisation of a Hamiltonian cycle in infinite graphs, or into a Hamiltonian circle and a spanning double-ray.

Original language | English |
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Pages (from-to) | 559-571 |

Number of pages | 13 |

Journal | Journal of Graph Theory |

Volume | 101 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 2022 |

## Keywords

- Alspach's conjecture
- Cayley graph
- Hamilton decomposition
- Hamiltonian cycle
- Hamiltonian decomposition
- cayley graph
- alspach's conjecture

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Geometry and Topology