论文标题
无限阿贝尔群的4个规范cayley图的汉密尔顿分解
Hamiltonian decompositions of 4-regular Cayley graphs of infinite abelian groups
论文作者
论文摘要
一个众所周知的Alspach的猜想说，Abelian群体的每2k $ groular cayley图可以分解为汉密尔顿周期。我们考虑了无限阿贝尔群体的一个类似问题。在这种环境中，哈密顿周期的一个自然类似物是跨越双射线。但是，由于存在有限的$ f $ f $ | f | $ | f | $和$ k $不同的$ 2K $常规的cayley图，因此对Alspach的猜想的幼稚概括无法在这种情况下保持。 我们表明，无限的Abelian组的每4美元的cayley图甚至可以将其有限切割分解为跨越双射线，因此当存在这种分解时，也可以表征。我们还表征了何时可以将这样的图分解为哈密顿圆圈，无限图中的哈密顿周期的拓扑概括，或者是哈密顿圆圈和跨越双射线的拓扑概括。
A well-known conjecture of Alspach says that every $2k$-regular Cayley graph of an 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 $2k$-regular Cayley graphs with finite cuts $F$ where $|F|$ and $k$ 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.
