Lax Lean Archive

The independence number is at most the annihilation number

Lax421170.IndependenceBound · concepts/Lax421170/IndependenceBound.lean · lax-421170

proven

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    Natural Language Statement

    Theorem

    For every finite simple graph GG,

    α(G)≤a(G),\alpha(G) \le a(G),

    where α(G)\alpha(G) is the independence number of GG, the largest number of pairwise nonadjacent vertices, and a(G)a(G) is the annihilation number of GG.

    In Lean, α(G)\alpha(G) is mathlib's SimpleGraph.indepNumSimpleGraph.indepNum, defined as the supremum of all kk for which GG has an independent set of exactly kk vertices.

    Concept map
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax421170.AnnihilationNumber
    2import Mathlib.Combinatorics.SimpleGraph.Clique
    3
    4/-!
    5---
    6title: The independence number is at most the annihilation number
    7type: theorem
    8---
    9For every finite simple graph GG,
    10
    11α(G)≤a(G),\alpha(G) \le a(G),
    12
    13where α(G)\alpha(G) is the independence number of GG, the largest number of
    14pairwise nonadjacent vertices, and a(G)a(G) is the annihilation number of GG.
    15
    16In Lean, α(G)\alpha(G) is mathlib's `SimpleGraph.indepNum`, defined as the
    17supremum of all kk for which GG has an independent set of exactly kk
    18vertices.
    19-/
    20
    21namespace Lax421170.IndependenceBound
    22
    23open Lax421170.AnnihilationNumber
    24
    25/-- The independence number of a finite simple graph is at most its annihilation number. -/
    26axiom indepNum_le_annihilationNumber {V : Type*} [Fintype V] (G : SimpleGraph V)
    27 [DecidableRel G.Adj] : G.indepNum ≤ annihilationNumber G
    28
    29end Lax421170.IndependenceBound
    30
    Show Proof
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    From Mathlib

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