Hamiltonian cycles in circulant digraphs with two stripes

The Circulant Travelling Salesman Problem (CTSP) is the problem of nding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. The computational complexity of this problem is not known. In fact, even the complexity of deciding Hamiltonicity of the underlying graph is unkown. This paper presents necessary and suucient conditions for the existence of a Hamilto-nian cycle in a digraph with circulant distance matrix consisting of only two stripes. These conditions can be checked in polynomial time. Moreover, a simple method for enumerating all Hamiltonian cycles in such a digraph is described. Based on these results we introduce two simple algorithms for solving the sum and bottleneck versions of CTSP for circulant distance matrices with two non-zero stripes.