We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.

翻译：我们研究了在基图对称性下哈密顿环的分类问题。重点聚焦于哈密顿传递图的极端情况，即图中每一对哈密顿环均存在一个图自同构将其相互映射的哈密顿图。这推广了已被广泛研究的唯一哈密顿图。本文证明，在温和条件及若干非平凡例外下，阿贝尔群的凯莱图不是哈密顿传递图，即它们至少包含两个结构不同的哈密顿环。为证明该结论，我们将哈密顿传递性归约到笛卡尔积分解的素因子性质上，这一归约方法本身具有独立研究价值。我们通过构造正则哈密顿传递图的无限族来补充结论，并构建了具有大量对称不等价哈密顿环的图族，从而考察了相反的极端情形。