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Degeneracy

Lax825442.Degeneracy · concepts/Lax825442/Degeneracy.lean · lax-825442

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    Natural Language Statement

    Definition

    Degeneracy is the least k such that every nonempty induced subgraph has a vertex of degree at most k.

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    1import Mathlib.Combinatorics.SimpleGraph.Acyclic
    2import Mathlib.Combinatorics.SimpleGraph.Finite
    3import Mathlib.Order.Lattice.Nat
    4
    5/-!
    6---
    7title: Degeneracy
    8type: definition
    9---
    10Degeneracy is the least k such that every nonempty induced subgraph has a
    11vertex of degree at most k.
    12-/
    13
    14namespace Lax825442.Degeneracy
    15
    16/-- Every nonempty induced subgraph has a vertex of degree at most `k`. -/
    17def HasDegeneracyAtMost {V : Type} [Fintype V] [DecidableEq V]
    18 (G : SimpleGraph V) (k : ℕ) : Prop :=
    19 ∀ S : Finset V, S.Nonempty →
    20 (∃ v : V, v ∈ S ∧
    21 ({u : V | u ∈ S ∧ G.Adj v u}.ncard ≤ k))
    22
    23/-- The least degeneracy bound. -/
    24noncomputable def degeneracy {V : Type} [Fintype V] [DecidableEq V]
    25 (G : SimpleGraph V) : ℕ :=
    26 sInf {k | HasDegeneracyAtMost G k}
    27
    28end Lax825442.Degeneracy
    29
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