Given a graph ▫$X$▫ with a Hamilton cycle ▫$C$▫, the {\em compression factor ▫$\kappa(X,C)$▫ of ▫$C$▫} is the order of the largest cyclic subgroup of ▫$\Aut(C)\cap\Aut(X)$▫, and the {\em Hamilton ...compression ▫$\kappa(X)$▫ of ▫$X$▫ } is the maximum of ▫$\kappa(X,C)$▫ where ▫$C$▫ runs over all Hamilton cycles in ▫$X$▫. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor, Merino and M\"utze in [``The Hamilton compression of highly symmetric graphs'', {\em arXiv preprint} \href{https://arxiv.org/abs/2205.08126}{arXiv: 2205.08126v1} (2022)] whether for every positive integer ▫$k$▫ there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to ▫$k$▫. Since an infinite family of Cayley graphs with Hamilton compression equal to ▫$1$▫ was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products ▫$\mathbb Z_p\rtimes\mathbb Z_k$▫ where ▫$p$▫ is a prime and ▫$k \geq 2$▫ a divisor of ▫$p-1$▫. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to ▫$1$▫ are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.
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