Lax Lean Archive

The independence number records the maximum cut

Lax762056.IndependenceIdentity · concepts/Lax762056/IndependenceIdentity.lean · lax-762056

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    For every finite simple graph FF, α(GF)=MaxCut⁡(F)+b(F)\alpha(G_F)=\operatorname{MaxCut}(F)+b(F). An independent set chooses at most one side in each column and so determines a cut of FF (empty columns may be assigned either side). The two rows for an edge uvuv contribute at most 2(v−u)+12(v-u)+1 vertices, and at most 2(v−u)2(v-u) when its endpoints lie on the same side. Conversely, every cut realizes the sum of these bounds.

    Concept map
    6 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    In the paper

    • page 5 of this submission's paper

    Lean source view on GitHub

    1import Lax762056.Reduction
    2
    3/-!
    4---
    5title: The independence number records the maximum cut
    6type: lemma
    7---
    8For every finite simple graph FF,
    9α(GF)=MaxCut⁡(F)+b(F)\alpha(G_F)=\operatorname{MaxCut}(F)+b(F).
    10An independent set chooses at most one side in each column and so determines
    11a cut of FF (empty columns may be assigned either side). The two rows for
    12an edge uvuv contribute at most 2(v−u)+12(v-u)+1 vertices, and at most 2(v−u)2(v-u)
    13when its endpoints lie on the same side. Conversely, every cut realizes
    14the sum of these bounds.
    15-/
    16
    17namespace Lax762056.IndependenceIdentity
    18
    19open Reduction MaxCut
    20
    21axiom independence_eq_maxCut_offset {n : ℕ} (F : SimpleGraph (Fin n)) :
    22 (reductionGraph F).indepNum = maxCut F + offset F
    23
    24end Lax762056.IndependenceIdentity
    25
    Show Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…