The Complexity of Hamiltonian Cycles and Paths with Prescribed Walk Lengths

Fix a set $S$ of positive integers. An $S$-Hamiltonian cycle is a cyclic ordering of all vertices of a graph such that each consecutive pair is joined by a walk whose length belongs to $S$. We study the complexity of deciding whether an input graph admits such an ordering, together with the corresponding path variants with prescribed or unrestricted endpoints. We complete the complexity classification for undirected graphs. In particular, we prove NP-completeness for every finite nonempty set of odd lengths, resolving an open problem posed by Amarilli, Lombardo, and Monet (WG 2026). For digraphs, all three problems are NP-complete for every fixed finite nonempty $S$ and admit linear-time algorithms when $S$ is cofinite. For other fixed infinite sets $S$, we establish NP-complete and undecidable cases, with undecidability occurring even when $S$ is recursive.