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Neighborhood diversity

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

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    Natural Language Statement

    Definition

    Neighborhood diversity is the minimum number of parts partitioning the vertices such that each part is a module and induces either a clique or an independent set. The empty graph has value zero.

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    1import Mathlib.Combinatorics.SimpleGraph.Finite
    2import Mathlib.Data.Set.Card
    3import Mathlib.Order.Lattice.Nat
    4
    5/-!
    6---
    7title: Neighborhood diversity
    8type: definition
    9---
    10Neighborhood diversity is the minimum number of parts partitioning the
    11vertices such that each part is a module and induces either a clique or an
    12independent set. The empty graph has value zero.
    13-/
    14
    15namespace Lax825442.NeighborhoodDiversity
    16
    17/-- A partition into `k` parts, each a module and each inducing a clique or
    18an independent set. Empty labels are allowed and disappear at the minimum. -/
    19def HasNeighborhoodDiversityAtMost {V : Type} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) (k : ℕ) : Prop :=
    21 ∃ part : V → Fin k,
    22 (∀ u v w : V,
    23 (part u = part v ∧ part w ≠ part u) → (G.Adj u w ↔ G.Adj v w)) ∧
    24 (∀ i : Fin k,
    25 (∀ u v : V,
    26 (u ≠ v ∧ (part u = i ∧ part v = i)) → G.Adj u v) ∨
    27 (∀ u v : V,
    28 (part u = i ∧ part v = i) → ¬ G.Adj u v))
    29
    30/-- The minimum number of neighborhood-diversity parts. -/
    31noncomputable def neighborhoodDiversity {V : Type} [Fintype V] [DecidableEq V]
    32 (G : SimpleGraph V) : ℕ :=
    33 sInf {k | HasNeighborhoodDiversityAtMost G k}
    34
    35end Lax825442.NeighborhoodDiversity
    36
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