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module
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Combinatorics.SimpleGraph.Matching@[expose] public sectionnamespace SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α]open Finset Listopen scoped Classical in
matchingNumber G is the size of a maximum matching of G.
noncomputable def matchingNumber (G : SimpleGraph α) [DecidableRel G.Adj] : ℝ :=
let matchings := { M : Subgraph G | M.IsMatching }
sSup (Set.image (fun M => (M.edgeSet.toFinset.card : ℝ)) matchings)In a finite graph, twice the number of edges of any matching is at most the number of vertices: each edge contributes two distinct vertices, and a matching's edges are vertex-disjoint.
α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αG:SimpleGraph αM:G.Subgraphinst✝:Fintype ↑M.edgeSeth:M.IsMatchinghdisj:∀ e ∈ M.edgeSet.toFinset, ∀ f ∈ M.edgeSet.toFinset, e ≠ f → _root_.Disjoint e.toFinset f.toFinsethcard2:∀ e ∈ M.edgeSet.toFinset, #e.toFinset = 2⊢ #(M.edgeSet.toFinset.biUnion Sym2.toFinset) ≤ #univ; All goals completed! 🐙end SimpleGraph