Lax Lean Archive

Lax49.GraphParameters

Graph parameters and functional equivalence

concepts/Lax49/GraphParameters.lean · lax-49

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    Definition

    A graph parameter assigns a natural number to every finite simple graph. Two graph parameters p and q are functionally equivalent when each is bounded by a numerical function of the other: there are functions f, g : ℕ → ℕ such that every finite simple graph G satisfies both p(G) ≤ f(q(G)) and q(G) ≤ g(p(G)). Functionally equivalent parameters are bounded on exactly the same classes of graphs.

    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Basic
    2
    3/-!
    4---
    5title: Graph parameters and functional equivalence
    6type: definition
    7---
    8A *graph parameter* assigns a natural number to every finite simple graph.
    9Two graph parameters *p* and *q* are *functionally equivalent* when each is
    10bounded by a numerical function of the other: there are functions
    11*f*, *g* : ℕ → ℕ such that every finite simple graph *G* satisfies both
    12*p*(*G*) ≤ *f*(*q*(*G*)) and *q*(*G*) ≤ *g*(*p*(*G*)). Functionally
    13equivalent parameters are bounded on exactly the same classes of graphs.
    14
    15# Formalization notes
    16
    17`GraphParam` is the uniform signature shared by all graph parameters in this
    18archive: a natural-valued function of finite simple graphs over `Fintype` and
    19`DecidableEq` instances. It is an abbreviation, so a statement about concrete
    20parameters mentions them by name — `Lax48.TwinWidth.twinWidth`,
    21`Lax49.MixedMinorNumber.mixedMinorNumber` — with no eta-expanded wrapper
    22lambdas a reader would have to unfold.
    23
    24The two directions of `FunctionallyEquivalent` are separate existential
    25claims rather than one function pair, so each can be established on its own.
    26-/
    27
    28namespace Lax49.GraphParameters
    29
    30/-- A natural-valued parameter of finite simple graphs, in the uniform
    31signature shared by all graph parameters in this archive. -/
    32abbrev GraphParam :=
    33 ∀ {V : Type} [Fintype V] [DecidableEq V], SimpleGraph V → ℕ
    34
    35/-- Each of two graph parameters is bounded by a numerical function of the
    36other. -/
    37def FunctionallyEquivalent (p q : GraphParam) : Prop :=
    38 (∃ f : ℕ → ℕ, ∀ {V : Type} [Fintype V] [DecidableEq V]
    39 (G : SimpleGraph V), p G ≤ f (q G)) ∧
    40 (∃ g : ℕ → ℕ, ∀ {V : Type} [Fintype V] [DecidableEq V]
    41 (G : SimpleGraph V), q G ≤ g (p G))
    42
    43end Lax49.GraphParameters
    44

    Formalization notes

    GraphParamGraphParam is the uniform signature shared by all graph parameters in this archive: a natural-valued function of finite simple graphs over FintypeFintype and DecidableEqDecidableEq instances. It is an abbreviation, so a statement about concrete parameters mentions them by name — Lax48.TwinWidth.twinWidthLax48.TwinWidth.twinWidth, Lax49.MixedMinorNumber.mixedMinorNumberLax49.MixedMinorNumber.mixedMinorNumber — with no eta-expanded wrapper lambdas a reader would have to unfold.

    The two directions of FunctionallyEquivalentFunctionallyEquivalent are separate existential claims rather than one function pair, so each can be established on its own.

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