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Cochromatic number

Lax560391.CochromaticNumber · concepts/Lax560391/CochromaticNumber.lean · lax-560391

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    Natural Language Statement

    Definition

    The cochromatic number is the least number of parts in a vertex partition where each part induces either a clique or an independent set. Parts need not be graph modules. The empty graph has cochromatic number zero.

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    DefinitionThis conceptRelated conceptA → B: B builds on A

    Lean source view on GitHub

    1import Lax560391.CochromaticPartition
    2import Mathlib.Order.Lattice.Nat
    3
    4/-!
    5---
    6title: Cochromatic number
    7type: definition
    8---
    9The cochromatic number is the least number of parts in a vertex partition
    10where each part induces either a clique or an independent set. Parts need not
    11be graph modules. The empty graph has cochromatic number zero.
    12-/
    13
    14namespace Lax560391.CochromaticNumber
    15
    16/-- A partition into at most `k` cliques or independent sets. Empty labels
    17are allowed and do not affect the minimum. -/
    18def HasCochromaticNumberAtMost {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop :=
    19 Nonempty (Lax560391.CochromaticPartition.CochromaticPartition G k)
    20
    21noncomputable def cochromaticNumber {V : Type} [Fintype V] [DecidableEq V]
    22 (G : SimpleGraph V) : ℕ :=
    23 sInf {k | HasCochromaticNumberAtMost G k}
    24
    25end Lax560391.CochromaticNumber
    26
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