Hamiltonian decompositions of 4-regular Cayley graphs of infinite abelian groups

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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 languageEnglish
Pages (from-to)559-571
Number of pages13
JournalJournal of Graph Theory
Volume101
Issue number3
DOIs
Publication statusAccepted/In press - 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

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