| import Mathlib.Combinatorics.SimpleGraph.Clique | |
| namespace OAI | |
| namespace HadwigerMatching | |
| universe u | |
| variable {V : Type u} | |
| /-- A finite matching whose edges are pairwise touching: each member is a | |
| two-vertex clique, and distinct members are disjoint and joined by an edge. -/ | |
| def IsTouchingMatching (G : SimpleGraph V) (M : Finset (Finset V)) : Prop := | |
| (∀ e ∈ M, e.card = 2 ∧ G.IsClique (e : Set V)) ∧ | |
| (∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f) ∧ | |
| (∀ e ∈ M, ∀ f ∈ M, e ≠ f → | |
| ∃ v ∈ e, ∃ w ∈ f, G.Adj v w) | |
| /-- The supremum of the cardinalities of pairwise touching matchings. | |
| For finite vertex types, this supremum is attained. -/ | |
| noncomputable def connectedMatchingNumber (G : SimpleGraph V) : ℕ := | |
| sSup {n : ℕ | ∃ M : Finset (Finset V), IsTouchingMatching G M ∧ M.card = n} | |
| end HadwigerMatching | |
| end OAI | |