A bstract Several counterexample models to the Nelson-Seiberg theorem have been discovered in previous literature, with generic superpotentials respecting R-symmetry and non-generic R-charge assignments for chiral fields. This work present a sufficient condition such models: number of 2 fields, which is greater than 0 must be less or equal fields plus independent field pairs opposite R-charges ...

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint theta graphs. As a corollary, every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint cycles of even length. 1. Terminology and introduction In this paper, we only co...