On the linear description of the 3-cycle polytope
Résumé
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 $PC^3_7$.