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Functional bounds between graph parameters

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

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    Natural Language Statement

    Definition

    For natural-valued graph parameters pp and qq, the arrow p→qp\to q means that there is a nondecreasing function f:N→Nf:\mathbb{N}\to\mathbb{N} such that q(G)≤f(p(G))q(G)\le f(p(G)) for every finite simple graph GG, including disconnected graphs. Thus a bound on pp gives a bound on qq.

    The parameter signature is the existing Lax153141.GraphParameters.GraphParamLax153141.GraphParameters.GraphParam.

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    Lean source view on GitHub

    1import Lax153141.GraphParameters
    2import Mathlib.Order.Monotone.Basic
    3
    4/-!
    5---
    6title: Functional bounds between graph parameters
    7type: definition
    8---
    9For natural-valued graph parameters pp and qq, the arrow p→qp\to q means
    10that there is a nondecreasing function f:N→Nf:\mathbb{N}\to\mathbb{N} such that
    11q(G)≤f(p(G))q(G)\le f(p(G)) for every finite simple graph GG, including disconnected
    12graphs. Thus a bound on pp gives a bound on qq.
    13
    14The parameter signature is the existing `Lax153141.GraphParameters.GraphParam`.
    15-/
    16
    17namespace Lax825442.Bounds
    18
    19open Lax153141.GraphParameters
    20
    21/-- A nondecreasing function of `p` bounds `q` on every finite simple graph. -/
    22def Bounds (p q : GraphParam) : Prop :=
    23 ∃ f : ℕ → ℕ, Monotone f ∧
    24 (∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
    25 q G ≤ f (p G))
    26
    27end Lax825442.Bounds
    28
    Builds on
    Used by
    From Mathlib

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