On Cycles that Alternate Through Selected Sets of Vertices

Given two disjoint sets of k vertices each in a graph G , it was recently asked whether δ ( G ) ≥ n / 2 and κ ( G ) ≥ 2 k would be a sufficient assumption to guarantee the existence of a Hamiltonian cycle of G which alternates visiting vertices of the two selected sets. We answer this question in the affirmative when the graph is large and we also provide a generalization to more sets.