Infinite Families of Vertex-Transitive Graphs with Prescribed Hamilton Compression

Abstract

Given a graph X with a Hamilton cycle C, the compression factor \(\kappa (X,C)\) of C is the order of the largest cyclic subgroup of \({\textrm{Aut}}\,(C)\cap {\textrm{Aut}}\,(X)\), and the 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 et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) 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 \(k \ge 2\) 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.