Saturation.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

The saturation number of a graph.

Main definitions

SimpleGraph.saturationNumber : the saturation number μ*(G), the minimum size of a maximal matching.
SimpleGraph.computableSaturationNumber : a computable companion definition.
SimpleGraph.saturationNumber_eq_computable : bridge equation saturationNumber G = computableSaturationNumber G.

Imports (4)

Imported by 2 files

def SimpleGraph.saturationNumber The saturation number μ*(G) (also called the lower matching number): the minimum number of edges in a maximal matching of G line 48 source
Source documentation
The saturation number μ*(G) (also called the lower matching number): the minimum number of edges in a maximal matching of G.
Every finite graph has a maximal matching (extend the empty matching greedily), so this set is nonempty and the infimum is a genuine minimum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, SimpleGraph, Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, Sym2, And, SimpleGraph.IsMaximalEdgeMatching, Eq, Finset.card
1 instances, coercions and auxiliary constants left out.
```
noncomputable def saturationNumber (G : SimpleGraph α) : ℕ :=
…
```
def SimpleGraph.computableSaturationNumber Computable saturation number via powerset enumeration over the edge set line 52 source
Source documentation
Computable saturation number via powerset enumeration over the edge set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Option.getD, Finset.min, Finset.image, Finset, Sym2, Finset.card, Finset.filter, SimpleGraph.IsMaximalEdgeMatching, Finset.powerset, SimpleGraph.edgeFinset, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
def computableSaturationNumber (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
theorem SimpleGraph.saturationNumber_eq_computable line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, Nat, SimpleGraph.saturationNumber, SimpleGraph.computableSaturationNumber
Proof uses: id, InfSet.sInf, Set.ofPred, Exists, Finset, Sym2, And, SimpleGraph.IsMaximalEdgeMatching, Finset.card, Set, Finset.image, Finset.filter, Finset.powerset, SimpleGraph.edgeFinset, Set.ext, Eq.mpr, Iff, Membership.mem, LE.le, congrArg and 33 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.