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module
public import Mathlib.Combinatorics.SimpleGraph.Clique
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Order.CompletePartialOrder@[expose] public sectionnamespace SimpleGraphvariable {V : Type*} {G : SimpleGraph V}open Finset List@[inherit_doc] scoped notation "α(" G ")" => indepNum G@[simp] lemma indepNum_of_isEmpty [IsEmpty V] : α(G) = 0 := V:Type u_1G:SimpleGraph Vinst✝:IsEmpty V⊢ α(G) = 0
All goals completed! 🐙V:Type u_1G:SimpleGraph Vinst✝¹:Finite Vinst✝:Nonempty Vval✝:Fintype Vv:V⊢ 0 + 1 ≤ α(G)
exact le_csSup ⟨Nat.card V, V:Type u_1G:SimpleGraph Vinst✝¹:Finite Vinst✝:Nonempty Vval✝:Fintype Vv:V⊢ Nat.card V ∈ upperBounds {n | ∃ s, G.IsNIndepSet n s} All goals completed! 🐙⟩
⟨{v}, V:Type u_1G:SimpleGraph Vinst✝¹:Finite Vinst✝:Nonempty Vval✝:Fintype Vv:V⊢ G.IsNIndepSet (0 + 1) {v} All goals completed! 🐙⟩
Infinite graphs: definitions for clique number so that the clique number of a graph
with unbounded clique size is ∞ rather than 0.
noncomputable
def ecliqueNum {V : Type} (G : SimpleGraph V) : ℕ∞ := ⨆ (s : Finset V) (_ : G.IsClique s), #s
The set of sizes of the cliques (maximal complete subgraphs) of G.
def cliqueSizes {V : Type} (G : SimpleGraph V) : Set ℕ :=
{ k | ∃ S : Finset V, Maximal (fun T : Finset V => G.IsClique (T : Set V)) S ∧ S.card = k }open scoped Classical in
The triangles of G containing the edge uv, that is, the 3-cliques of G whose vertex
set contains both endpoints of uv.
noncomputable def trianglesContaining {α : Type*} [Fintype α] (G : SimpleGraph α)
(uv : Sym2 α) : Finset (Finset α) :=
(G.cliqueFinset 3).filter (fun t ↦ uv.toFinset ⊆ t)end SimpleGraph