On the linear description of the k-cycle polytope

We study a linear description of PC^kn the convex hull of incidence vectors of all the cycles consisting of exactly k edges (k ⩾ 4) in Kn, the complete undirected graph with n vertices. First, we describe some properties of PC^kn. Second, we discuss relations between PC^kn and PC^k′n′ with k′ > k and n′ > n. Then we give three lifting algorithms that transform a facet of PC^kn into facets of PC^kn′ with n′ > n. Finally, we provide an integer formulation and a partial linear description of PC^kn.