Chorded cycle facets of the clique partitioning polytope

The q -chorded k -cycle inequalities are a class of valid inequalities for the clique partitioning polytope. It is known that for q ∈ { 2 , 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 = 3 q + 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.