FormalConjecturesForMathlib.Combinatorics.SimpleGraph.Degeneracy

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Copyright 2026 The Formal Conjectures Authors.

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you may not use this file except in compliance with the License.
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-/

module

public import Mathlib.Combinatorics.SimpleGraph.Basic
public import Mathlib.Data.Finset.Card

@[expose] public section

Graph degeneracy

A graph is $r$-degenerate if each nonempty finite vertex set contains a vertex with at most $r$ neighbours in that set.

namespace SimpleGraphopen scoped Classical in

The neighbours of v lying inside s.

noncomputable def neighborsWithin {V : Type*} (H : SimpleGraph V) (s : Finset V) (v : V) :
    Finset V := s.filter (H.Adj v)

H is r-degenerate when every induced subgraph has a vertex of degree at most r, that is, every nonempty vertex set contains a vertex with at most r neighbours inside it.

def IsDegenerate {V : Type*} (H : SimpleGraph V) (r : ℕ) : Prop :=
  ∀ s : Finset V, s.Nonempty → ∃ v ∈ s, (neighborsWithin H s v).card ≤ r

@[simp]
lemma neighborsWithin_empty {V : Type*} (H : SimpleGraph V) (v : V) :
    H.neighborsWithin ∅ v = ∅ := byV:Type u_1H:SimpleGraph Vv:V⊢ H.neighborsWithin ∅ v = ∅
  simp [neighborsWithin]All goals completed! 🐙

@[simp]
lemma isDegenerate_bot {V : Type*} (r : ℕ) : (⊥ : SimpleGraph V).IsDegenerate r := byV:Type u_1r:ℕ⊢ ⊥.IsDegenerate r
  intro s hsV:Type u_1r:ℕs:Finset Vhs:s.Nonempty⊢ ∃ v ∈ s, (⊥.neighborsWithin s v).card ≤ r
  obtain ⟨v, hv⟩ := hsV:Type u_1r:ℕs:Finset Vv:Vhv:v ∈ s⊢ ∃ v ∈ s, (⊥.neighborsWithin s v).card ≤ r
  exact ⟨v, hv, byV:Type u_1r:ℕs:Finset Vv:Vhv:v ∈ s⊢ (⊥.neighborsWithin s v).card ≤ r simp [neighborsWithin]All goals completed! 🐙⟩

end SimpleGraph