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 $e\in E(H)$ such that $e\cap M\ne \emptyset$ and $e\setminus M\ne \emptyset$, there exists $m\in M$ such that $e\cap M=\{m\}$ and for every $n\in M$, we have $(e\setminus \{m\})\cup \{n\}\in E(H)$. For example, $\emptyset$, V(H) and $\{v\}$, where $v\in V(H)$, 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 $|V(H)|\ge 4$, there exist $v,w\in V(H)$ such that $H-\{v,w\}$ is prime.