On the linear description of the 3-cycle polytope

Let Kn be the complete undirected graph with n vertices. A 3-cycle is a simple cycle consisting of exactly 3-edges in Kn. The 3-cycle polytope, PC³n is defined as the convex hull of the incidence vectors of all 3-cycles in Kn. This polytope is proved to be a facet of the dominant of all cycles in Kn. In this paper, we design some sequential lifting procedures for facet-defining inequalities of PC³n. Using these lifting procedures, we show that the number of distinct coefficients in facet-defining inequalities of PC³n increases strictly when n grows and the maximum difference between the greatest coefficient and the smallest coefficient in some facet-defining inequalities is exponential in n. We also discuss relationships between PC³n and other ‘cycle’ polytopes. Finally in A we give a complete description of PC³8 and show that all of its facets except one can be lifted from the facets of PC³5, of PC³6 and PC³7.