Chordal.lean`

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9 declarations, 0 contain sorry, 0 declared as axioms

Chordal graphs and clique edge partitions

Definitions and basic results used in Erdős Problem 81.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.IsChordal A graph is chordal if it contains no induced cycle of length k ≥ 4 line 35 source
Source documentation
A graph is chordal if it contains no induced cycle of length k ≥ 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: Nat, LE.le, OfNat.ofNat, Not, SimpleGraph.IsIndContained, Fin, SimpleGraph.cycleGraph
2 instances, coercions and auxiliary constants left out.
```
def IsChordal (G : SimpleGraph V) : Prop :=
…
```
def SimpleGraph.IsCliqueEdgePartition Documented declaration: IsCliqueEdgePartition line 41 source
Source documentation
P is a partition of the edges of G into cliques: each member of P is a clique of G with at least two vertices, and every edge of G is contained in exactly one member of P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset
Proof uses: And, Membership.mem, LE.le, Nat, OfNat.ofNat, Finset.card, SimpleGraph.IsClique, SimpleGraph.Adj, ExistsUnique
5 instances, coercions and auxiliary constants left out.
```
def IsCliqueEdgePartition (G : SimpleGraph V) (P : Finset (Finset V)) : Prop :=
…
```
def SimpleGraph.IsSplitGraph A graph is a split graph if its vertex set splits into a clique and an independent set line 47 source
Source documentation
A graph is a split graph if its vertex set splits into a clique and an independent set.

no axioms what this means
Statement uses: SimpleGraph
Proof uses: Exists, Set, And, SimpleGraph.IsClique, SimpleGraph.IsIndepSet, Compl.compl
1 instances, coercions and auxiliary constants left out.
```
def IsSplitGraph (G : SimpleGraph V) : Prop :=
…
```
theorem SimpleGraph.exists_isCliqueEdgePartition_card_eq Every finite graph has a clique edge partition consisting of its edges line 51 source
Source documentation
Every finite graph has a clique edge partition consisting of its edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Exists, Finset, And, SimpleGraph.IsCliqueEdgePartition, Eq, Nat, Finset.card, Sym2, SimpleGraph.edgeFinset
Proof uses: Exists.intro, Finset.image, Sym2.toFinset, And.intro, Membership.mem, LE.le, OfNat.ofNat, SimpleGraph.IsClique, ExistsUnique, Eq.mpr, Set, SimpleGraph.edgeSet, id, forall_congr, implies_congr, Eq.trans, congrArg, funext, congrFun', Eq.refl and 28 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
```
theorem exists_isCliqueEdgePartition_card_eq {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj] :
    ∃ P : Finset (Finset V), G.IsCliqueEdgePartition P ∧ P.card = G.edgeFinset.card := by
…
```
lemma SimpleGraph.cycleGraph_adj_succ line 80 source
lemma SimpleGraph.cycleGraph_not_adj_add_two line 83 source
theorem SimpleGraph.IsSplitGraph.isChordal Every split graph is chordal line 101 source
Source documentation
Every split graph is chordal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, SimpleGraph.IsSplitGraph, SimpleGraph.IsChordal
Proof uses: Set, And, SimpleGraph.IsClique, SimpleGraph.IsIndepSet, Compl.compl, Nat, LE.le, OfNat.ofNat, SimpleGraph.IsIndContained, Fin, SimpleGraph.cycleGraph, False, SimpleGraph.Embedding, Eq, HAdd.hAdd, Exists, Nat.exists_eq_add_of_le', Membership.mem, SimpleGraph.Adj, Ne and 12 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
theorem IsSplitGraph.isChordal {V : Type*} {G : SimpleGraph V} (hG : G.IsSplitGraph) :
    G.IsChordal := by
…
```
theorem SimpleGraph.isCliqueEdgePartition_bot_empty The empty family partitions the edges of an edgeless graph line 122 source
Source documentation
The empty family partitions the edges of an edgeless graph.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph.IsCliqueEdgePartition, Bot.bot, SimpleGraph, EmptyCollection.emptyCollection, Finset
Proof uses: of_eq_true, And, Membership.mem, LE.le, Nat, OfNat.ofNat, Finset.card, SimpleGraph.IsClique, SimpleGraph.Adj, ExistsUnique, Eq.trans, True, congr, congrArg, forall_congr, False, Set.Subsingleton, implies_congr, implies_true, funext and 3 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
@[simp] theorem isCliqueEdgePartition_bot_empty :
    (⊥ : SimpleGraph V).IsCliqueEdgePartition ∅ := by
…
```
Mentions: SimpleGraph.IsCliqueEdgePartition
theorem SimpleGraph.isSplitGraph_bot An edgeless graph is split, including when its vertex type is empty line 127 source
Source documentation
An edgeless graph is split, including when its vertex type is empty.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph.IsSplitGraph, Bot.bot, SimpleGraph
Proof uses: Exists.intro, Set, And, SimpleGraph.IsClique, SimpleGraph.IsIndepSet, Compl.compl, EmptyCollection.emptyCollection, And.intro, of_eq_true, Eq.trans, Set.Subsingleton, True, Membership.mem, Not, Eq, SimpleGraph.Adj, forall_congr, implies_congr, Set.univ, congrFun' and 6 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem isSplitGraph_bot : (⊥ : SimpleGraph V).IsSplitGraph := by
…
```
Mentions: SimpleGraph.IsSplitGraph