Spanning Structures in Multipartite Graph Traversals

Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. We consider the problem of determining how large $\alpha$ must be in order to guarantee that $G$ has a Hamiltonian traversal (an $r$-cycle subgraph containing exactly one vertex from each part), and show that this critical density tends to $\frac{1}{2}$ as $r$ increases. This resolves a conjecture of Badakhshian, Falgas-Ravry, and Sharifzadeh. We also study the critical densities necessary to guarantee the existence of other spanning structures in traversals, particularly subgraph factors, and obtain asymptotically the critical densities for traversal $F$-factor subgraphs for several classes of graphs $F$. The proofs of our results involve the absorption method.