Hamiltonian Cycles in Striped Graphs: The Two-Stripe Problem

For a directed graph $G = (N,A)$, the kth stripe of G is $R_k = \{ (i,[i + k]_n )\} $ where $[a]_n $ is a (modulo n) and n is the number of nodes. A graph is striped if A consists of a set of stripes. The t-stripe problem is to determine whether a graph containing t stripes is hamiltonian. Necessary and sufficient conditions are provided for $t\leqq 2$, although the problem is still open for $t\geqq 3$.