On the linear description of the 3-cycle polytope

Abstract Let K n be the complete undirected graph with n vertices. A 3-cycle is a simple cycle consisting of exactly 3-edges in K n . The 3-cycle polytope, PC 3 n is defined as the convex hull of the incidence vectors of all 3-cycles in K n . This polytope is proved to be a facet of the dominant of all cycles in K n [Math. Oper. Res. 22 (1997) 110]. In this paper, we design some sequential lifting procedures for facet-defining inequalities of PC 3 n . Using these lifting procedures, we show that the number of distinct coefficients in facet-defining inequalities of PC 3 n 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 PC 3 n and other “cycle” polytopes. Finally in Appendix A we give a complete description of PC 3 8 and show that all of its facets except one can be lifted from the facets of PC 3 5 , of PC 3 6 and PC 3 7 .