The $q$-chorded $k$-cycle inequalities are a class of valid inequalities for the clique partitioning polytope. It is known that for $q = 2$ or $q = \tfrac{k-1}{2}$, these inequalities induce facets of the clique partitioning polytope if and only if $k$ is odd. We solve the open problem of characterizing 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 $k$-循环不等式是团划分多面体的一类有效不等式。已知当 $q = 2$ 或 $q = \tfrac{k-1}{2}$ 时，这些不等式诱导出团划分多面体的面当且仅当 $k$ 为奇数。我们解决了对任意 $k$ 和 $q$ 刻画此类面的公开问题。具体而言，我们证明 $q$-弦 $k$-循环不等式诱导出团划分多面体的面当且仅当满足两个条件：$k = 1 \mod q$，且若 $k=3q+1$ 则 $q=3$ 或 $q$ 为偶数。这一结论确立了除先前已知情形外，由 $q$-弦 $k$-循环不等式诱导的众多面的存在性。