The $q$-chorded $k$-cycle inequalities are a class of valid inequalities for the clique partitioning polytope. It is known that for $q \in \{2, \tfrac{k-1}{2}\}$, these inequalities induce facets of the clique partitioning polytope if and only if $k$ is odd. Here, we characterize such facets for arbitrary $k$ and $q$. More specifically, we prove that the $q$-chorded $k$-cycle inequalities induce facets of the clique partitioning polytope if and only if two conditions are satisfied: $k = 1$ mod $q$, and if $k=3q+1$ then $q=3$ or $q$ is even. This establishes the existence of many facets induced by $q$-chorded $k$-cycle inequalities beyond those previously known.

翻译：$q$-弦$k$-环不等式是团划分多面体的一类有效不等式。已知当$q \in \{2, \tfrac{k-1}{2}\}$时，这些不等式诱导团划分多面体刻面的充要条件是$k$为奇数。本文刻画了任意$k$与$q$情形下此类刻面的特征。具体而言，我们证明$q$-弦$k$-环不等式诱导团划分多面体刻面的充要条件是满足以下两个条件：$k = 1$ mod $q$，且当$k=3q+1$时需满足$q=3$或$q$为偶数。该结论确立了超出既往认知的、由$q$-弦$k$-环不等式诱导的众多刻面的存在性。