Prime 3-uniform hypergraphs

Pierre Ille (1) , Abderrahim Boussaïri , Brahim Chergui , Mohamed Zaidi

1 I2M - Institut de Mathématiques de Marseille

Given a 3-uniform hypergraph H, a subset M of V (H) is a module of H if for each e ∈ E(H) such that e ∩ M ≠ ∅ and e ∖ M ≠ ∅, there exists m ∈ M such that e ∩ M = {m} and for every n ∈ M , we have (e ∖ {m}) ∪ {n} ∈ E(H). For example, ∅, V (H) and {v}, where 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) ≥ 4, there exist v, w ∈ V (H) such that H − {v, w} is prime.

Mots clés: 3-uniform; module; prime

Domaines: Combinatoire [math.CO]

Pierre Ille, Abderrahim Boussaïri, Brahim Chergui, Mohamed Zaidi. Prime 3-uniform hypergraphs. Graphs and Combinatorics, 2021, 37, pp.2737-2760. ⟨hal-03362671⟩