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Ordinary fractional coloring

Lax342547.FractionalColoring · concepts/Lax342547/FractionalColoring.lean · lax-342547

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    Natural Language Statement

    Lemma

    A fractional coloring assigns nonnegative real weights to independent sets so that the weights covering each vertex sum to at least one. The fractional chromatic number χf(G)\chi_f(G) is the infimum of the total weight. For finite graphs with independence number at most two, ∣V(G)∣/2≤χf(G)≤χ(G)|V(G)|/2\le\chi_f(G)\le\chi(G).

    Concept map
    1 concept; 1 descendant hidden
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 fractional_chromatic_le_chromatic proven

    2 fractional_chromatic_lower_bound proven

    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
    2import Mathlib.Data.Fintype.Powerset
    3import Mathlib.Algebra.Order.Archimedean.Real.Hom
    4
    5/-!
    6---
    7title: Ordinary fractional coloring
    8type: lemma
    9---
    10A fractional coloring assigns nonnegative real weights to independent sets
    11so that the weights covering each vertex sum to at least one. The fractional
    12chromatic number χf(G)\chi_f(G) is the infimum of the total weight.
    13For finite graphs with independence number at most two,
    14∣V(G)∣/2≤χf(G)≤χ(G)|V(G)|/2\le\chi_f(G)\le\chi(G).
    15-/
    16
    17namespace Lax342547.FractionalColoring
    18
    19open scoped BigOperators
    20
    21universe u
    22variable {V : Type u} [Fintype V] [DecidableEq V]
    23
    24structure Coloring (G : SimpleGraph V) where
    25 weight : Finset V → ℝ
    26 nonneg : ∀ S, 0 ≤ weight S
    27 supported : ∀ (S : Finset V), ¬ G.IsIndepSet (S : Set V) → weight S = 0
    28 covers : ∀ v, 1 ≤ ∑ S : Finset V, if v ∈ S then weight S else 0
    29
    30noncomputable def totalWeight {G : SimpleGraph V} (C : Coloring G) : ℝ :=
    31 ∑ S : Finset V, C.weight S
    32
    33noncomputable def fractionalChromaticNumber (G : SimpleGraph V) : ℝ :=
    34 sInf {w : ℝ | ∃ C : Coloring G, totalWeight C = w}
    35
    36axiom fractional_chromatic_lower_bound (G : SimpleGraph V) (hα : G.indepNum ≤ 2) :
    37 (Fintype.card V : ℝ) / 2 ≤ fractionalChromaticNumber G
    38
    39axiom fractional_chromatic_le_chromatic (G : SimpleGraph V) :
    40 fractionalChromaticNumber G ≤ (G.chromaticNumber.toNat : ℝ)
    41
    42end Lax342547.FractionalColoring
    43
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