Graph cliques

This file defines cliques in simple graphs. A clique is a set of vertices that are pairwise adjacent.

Main declarations

• SimpleGraph.IsClique: Predicate for a set of vertices to be a clique.
• SimpleGraph.IsNClique: Predicate for a set of vertices to be an n-clique.
• SimpleGraph.cliqueFinset: Finset of n-cliques of a graph.
• SimpleGraph.CliqueFree: Predicate for a graph to have no n-cliques.

TODO

• Clique numbers
• Do we need cliqueSet, a version of cliqueFinset for infinite graphs?

Cliques

@[inline, reducible]

abbrev SimpleGraph.IsClique {α : Type u_1} (G : SimpleGraph α) (s : Set α) : Prop

A clique in a graph is a set of vertices that are pairwise adjacent.

theorem SimpleGraph.isClique_iff {α : Type u_1} (G : SimpleGraph α) {s : Set α} : SimpleGraph.IsClique G s ↔ Set.Pairwise s G.Adj

theorem SimpleGraph.isClique_iff_induce_eq {α : Type u_1} (G : SimpleGraph α) {s : Set α} : SimpleGraph.IsClique G s ↔ SimpleGraph.induce s G = ⊤

A clique is a set of vertices whose induced graph is complete.

instance SimpleGraph.instDecidableIsCliqueToSet {α : Type u_1} (G : SimpleGraph α) [DecidableEq α] [DecidableRel G.Adj] {s : Finset α} : Decidable (SimpleGraph.IsClique G ↑s)

theorem SimpleGraph.IsClique.mono {α : Type u_1} {G : SimpleGraph α} {H : SimpleGraph α} {s : Set α} (h : G ≤ H) : SimpleGraph.IsClique G s → SimpleGraph.IsClique H s

theorem SimpleGraph.IsClique.subset {α : Type u_1} {G : SimpleGraph α} {s : Set α} {t : Set α} (h : t ⊆ s) : SimpleGraph.IsClique G s → SimpleGraph.IsClique G t

@[simp]

theorem SimpleGraph.isClique_bot_iff {α : Type u_1} {s : Set α} : SimpleGraph.IsClique ⊥ s ↔ Set.Subsingleton s

theorem SimpleGraph.IsClique.subsingleton {α : Type u_1} {s : Set α} : SimpleGraph.IsClique ⊥ s → Set.Subsingleton s

Alias of the forward direction of SimpleGraph.isClique_bot_iff.

n-cliques

structure SimpleGraph.IsNClique {α : Type u_1} (G : SimpleGraph α) (n : ℕ) (s : Finset α) : Prop

• clique : SimpleGraph.IsClique G ↑s✝
• card_eq : Finset.card s✝ = n

An n-clique in a graph is a set of n vertices which are pairwise connected.

theorem SimpleGraph.isNClique_iff {α : Type u_1} (G : SimpleGraph α) {n : ℕ} {s : Finset α} : SimpleGraph.IsNClique G n s ↔ SimpleGraph.IsClique G ↑s ∧ Finset.card s = n

instance SimpleGraph.instDecidableIsNClique {α : Type u_1} (G : SimpleGraph α) [DecidableEq α] [DecidableRel G.Adj] {n : ℕ} {s : Finset α} : Decidable (SimpleGraph.IsNClique G n s)

theorem SimpleGraph.IsNClique.mono {α : Type u_1} {G : SimpleGraph α} {H : SimpleGraph α} {n : ℕ} {s : Finset α} (h : G ≤ H) : SimpleGraph.IsNClique G n s → SimpleGraph.IsNClique H n s

@[simp]

theorem SimpleGraph.isNClique_bot_iff {α : Type u_1} {n : ℕ} {s : Finset α} : SimpleGraph.IsNClique ⊥ n s ↔ n ≤ 1 ∧ Finset.card s = n

theorem SimpleGraph.is3Clique_triple_iff {α : Type u_1} {G : SimpleGraph α} [DecidableEq α] {a : α} {b : α} {c : α} : SimpleGraph.IsNClique G 3 {a, b, c} ↔ SimpleGraph.Adj G a b ∧ SimpleGraph.Adj G a c ∧ SimpleGraph.Adj G b c

theorem SimpleGraph.is3Clique_iff {α : Type u_1} {G : SimpleGraph α} {s : Finset α} [DecidableEq α] : SimpleGraph.IsNClique G 3 s ↔ ∃ a b c, SimpleGraph.Adj G a b ∧ SimpleGraph.Adj G a c ∧ SimpleGraph.Adj G b c ∧ s = {a, b, c}

Graphs without cliques

def SimpleGraph.CliqueFree {α : Type u_1} (G : SimpleGraph α) (n : ℕ) : Prop

G.CliqueFree n means that G has no n-cliques.

theorem SimpleGraph.IsNClique.not_cliqueFree {α : Type u_1} {G : SimpleGraph α} {n : ℕ} {s : Finset α} (hG : SimpleGraph.IsNClique G n s) : ¬SimpleGraph.CliqueFree G n

theorem SimpleGraph.not_cliqueFree_of_top_embedding {α : Type u_1} {G : SimpleGraph α} {n : ℕ} (f : ⊤ ↪g G) : ¬SimpleGraph.CliqueFree G n

noncomputable def SimpleGraph.topEmbeddingOfNotCliqueFree {α : Type u_1} {G : SimpleGraph α} {n : ℕ} (h : ¬SimpleGraph.CliqueFree G n) : ⊤ ↪g G

An embedding of a complete graph that witnesses the fact that the graph is not clique-free.

theorem SimpleGraph.not_cliqueFree_iff {α : Type u_1} {G : SimpleGraph α} (n : ℕ) : ¬SimpleGraph.CliqueFree G n ↔ Nonempty (⊤ ↪g G)

theorem SimpleGraph.cliqueFree_iff {α : Type u_1} {G : SimpleGraph α} {n : ℕ} : SimpleGraph.CliqueFree G n ↔ IsEmpty (⊤ ↪g G)

theorem SimpleGraph.not_cliqueFree_card_of_top_embedding {α : Type u_1} {G : SimpleGraph α} [Fintype α] (f : ⊤ ↪g G) : ¬SimpleGraph.CliqueFree G (Fintype.card α)

theorem SimpleGraph.cliqueFree_bot {α : Type u_1} {n : ℕ} (h : 2 ≤ n) : SimpleGraph.CliqueFree ⊥ n

theorem SimpleGraph.CliqueFree.mono {α : Type u_1} {G : SimpleGraph α} {m : ℕ} {n : ℕ} (h : m ≤ n) : SimpleGraph.CliqueFree G m → SimpleGraph.CliqueFree G n

theorem SimpleGraph.CliqueFree.anti {α : Type u_1} {G : SimpleGraph α} {H : SimpleGraph α} {n : ℕ} (h : G ≤ H) : SimpleGraph.CliqueFree H n → SimpleGraph.CliqueFree G n

theorem SimpleGraph.cliqueFree_of_card_lt {α : Type u_1} {G : SimpleGraph α} {n : ℕ} [Fintype α] (hc : Fintype.card α < n) : SimpleGraph.CliqueFree G n

See SimpleGraph.cliqueFree_of_chromaticNumber_lt for a tighter bound.

theorem SimpleGraph.cliqueFree_completeMultipartiteGraph {n : ℕ} {ι : Type u_2} [Fintype ι] (V : ι → Type u_3) (hc : Fintype.card ι < n) : SimpleGraph.CliqueFree (SimpleGraph.completeMultipartiteGraph V) n

A complete r-partite graph has no n-cliques for r < n.

Set of cliques

def SimpleGraph.cliqueSet {α : Type u_1} (G : SimpleGraph α) (n : ℕ) : Set (Finset α)

The n-cliques in a graph as a set.

theorem SimpleGraph.mem_cliqueSet_iff {α : Type u_1} (G : SimpleGraph α) {n : ℕ} {s : Finset α} : s ∈ SimpleGraph.cliqueSet G n ↔ SimpleGraph.IsNClique G n s

@[simp]

theorem SimpleGraph.cliqueSet_eq_empty_iff {α : Type u_1} (G : SimpleGraph α) {n : ℕ} : SimpleGraph.cliqueSet G n = ∅ ↔ SimpleGraph.CliqueFree G n

theorem SimpleGraph.CliqueFree.cliqueSet {α : Type u_1} (G : SimpleGraph α) {n : ℕ} : SimpleGraph.CliqueFree G n → SimpleGraph.cliqueSet G n = ∅

Alias of the reverse direction of SimpleGraph.cliqueSet_eq_empty_iff.

theorem SimpleGraph.cliqueSet_mono {α : Type u_1} {G : SimpleGraph α} {H : SimpleGraph α} {n : ℕ} (h : G ≤ H) : SimpleGraph.cliqueSet G n ⊆ SimpleGraph.cliqueSet H n

theorem SimpleGraph.cliqueSet_mono' {α : Type u_1} {G : SimpleGraph α} {H : SimpleGraph α} (h : G ≤ H) : SimpleGraph.cliqueSet G ≤ SimpleGraph.cliqueSet H

Finset of cliques

def SimpleGraph.cliqueFinset {α : Type u_1} (G : SimpleGraph α) [Fintype α] [DecidableEq α] [DecidableRel G.Adj] (n : ℕ) : Finset (Finset α)

The n-cliques in a graph as a finset.

theorem SimpleGraph.mem_cliqueFinset_iff {α : Type u_1} (G : SimpleGraph α) [Fintype α] [DecidableEq α] [DecidableRel G.Adj] {n : ℕ} {s : Finset α} : s ∈ SimpleGraph.cliqueFinset G n ↔ SimpleGraph.IsNClique G n s

@[simp]

theorem SimpleGraph.coe_cliqueFinset {α : Type u_1} (G : SimpleGraph α) [Fintype α] [DecidableEq α] [DecidableRel G.Adj] (n : ℕ) : ↑(SimpleGraph.cliqueFinset G n) = SimpleGraph.cliqueSet G n

@[simp]

theorem SimpleGraph.cliqueFinset_eq_empty_iff {α : Type u_1} (G : SimpleGraph α) [Fintype α] [DecidableEq α] [DecidableRel G.Adj] {n : ℕ} : SimpleGraph.cliqueFinset G n = ∅ ↔ SimpleGraph.CliqueFree G n

theorem SimpleGraph.CliqueFree.cliqueFinset {α : Type u_1} (G : SimpleGraph α) [Fintype α] [DecidableEq α] [DecidableRel G.Adj] {n : ℕ} : SimpleGraph.CliqueFree G n → SimpleGraph.cliqueFinset G n = ∅

Alias of the reverse direction of SimpleGraph.cliqueFinset_eq_empty_iff.

theorem SimpleGraph.cliqueFinset_mono {α : Type u_1} {G : SimpleGraph α} (H : SimpleGraph α) [Fintype α] [DecidableEq α] [DecidableRel G.Adj] {n : ℕ} [DecidableRel H.Adj] (h : G ≤ H) : SimpleGraph.cliqueFinset G n ⊆ SimpleGraph.cliqueFinset H n