Lax Lean Archive

χ-Boundedness

Lax9.ChiBoundedness · concepts/Lax9/ChiBoundedness.lean · lax-9

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    Natural Language Statement

    Definition

    A class of finite simple graphs is χ-bounded if the chromatic number of each graph in the class is bounded by a function of its clique number.

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    1import Lax9.MergeWidth
    2
    3/-!
    4---
    5title: χ-Boundedness
    6type: definition
    7---
    8A class of finite simple graphs is χ-bounded if the chromatic number of each
    9graph in the class is bounded by a function of its clique number.
    10-/
    11
    12namespace Lax9.ChiBoundedness
    13
    14open Lax9.MergeWidth
    15
    16/-- A graph class CC is χ-bounded if there is a function ff such that every
    17G∈CG ∈ C is f(ω(G))f(ω(G))-colourable. -/
    18def ChiBounded (C : GraphClass) : Prop :=
    19 ∃ f : ℕ → ℕ, ∀ ⦃V : Type⦄ [Fintype V] (G : SimpleGraph V), C G →
    20 G.Colorable (f G.cliqueNum)
    21
    22end Lax9.ChiBoundedness
    23
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