Prime 3-Uniform Hypergraphs

Abstract

Given a 3-uniform hypergraph H, a subset M of V(H) is a module of H if for each $$e\in E(H)$$ e ∈ E ( H ) such that $$e\cap M\ne \emptyset$$ e ∩ M ≠ ∅ and $$e\setminus M\ne \emptyset$$ e \ M ≠ ∅ , there exists $$m\in M$$ m ∈ M such that $$e\cap M=\{m\}$$ e ∩ M = { m } and for every $$n\in M$$ n ∈ M , we have $$(e\setminus \{m\})\cup \{n\}\in E(H)$$ ( e \ { m } ) ∪ { n } ∈ E ( H ) . For example, $$\emptyset$$ ∅ , V(H) and $$\{v\}$$ { v } , where $$v\in V(H)$$ v ∈ V ( H ) , are modules of H, called trivial. A 3-uniform hypergraph is prime if all its modules are trivial. Given a prime 3-uniform hypergraph, we study its prime, 3-uniform and induced subhypergraphs. Our main result is: given a prime 3-uniform hypergraph H, with $$|V(H)|\ge 4$$ | V ( H ) | ≥ 4 , there exist $$v,w\in V(H)$$ v , w ∈ V ( H ) such that $$H-\{v,w\}$$ H - { v , w } is prime.