Lax Lean Archive

Bicliques with pendant neighbors

Lax689614.Biclique · concepts/Lax689614/Biclique.lean · lax-689614

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    Natural Language Statement

    Theorem

    Lemma 6. Partition the surviving vertices into sets L,R,IL,R,I. The sets LL and RR induce a complete bipartite graph, II is independent, and each vertex of L∪RL\cup R has a neighbor in II whose only surviving neighbor is that vertex. Other edges between L∪RL\cup R and II are unrestricted. The value is g(∣L∣,∣R∣)=(∣L∣+∣R∣) mod 2+2(min⁡(∣L∣,∣R∣) mod 2)g(|L|,|R|)=(|L|+|R|)\bmod 2+2(\min(|L|,|R|)\bmod 2). Either side of the biclique may be empty.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Lax689614.Grundy
    2
    3/-!
    4---
    5title: Bicliques with pendant neighbors
    6type: theorem
    7---
    8Lemma 6. Partition the surviving vertices into sets L,R,IL,R,I. The sets
    9LL and RR induce a complete bipartite graph, II is independent, and
    10each vertex of L∪RL\cup R has a neighbor in II whose only surviving
    11neighbor is that vertex. Other edges between L∪RL\cup R and II are
    12unrestricted. The value is
    13g(∣L∣,∣R∣)=(∣L∣+∣R∣) mod 2+2(min⁡(∣L∣,∣R∣) mod 2)g(|L|,|R|)=(|L|+|R|)\bmod 2+2(\min(|L|,|R|)\bmod 2).
    14Either side of the biclique may be empty.
    15-/
    16
    17namespace Lax689614.Biclique
    18
    19def g (a b : ℕ) : ℕ := (a + b) % 2 + 2 * (min a b % 2)
    20
    21structure Partition {V : Type} [DecidableEq V] (G : SimpleGraph V)
    22 (S L R I : Finset V) : Prop where
    23 cover : S = L ∪ R ∪ I
    24 left_right : Disjoint L R
    25 left_independent : Disjoint L I
    26 right_independent : Disjoint R I
    27 left_stable : ∀ u ∈ L, ∀ v ∈ L, ¬ G.Adj u v
    28 right_stable : ∀ u ∈ R, ∀ v ∈ R, ¬ G.Adj u v
    29 independent_stable : ∀ u ∈ I, ∀ v ∈ I, ¬ G.Adj u v
    30 complete : ∀ u ∈ L, ∀ v ∈ R, G.Adj u v
    31 pendant : ∀ u ∈ L ∪ R, ∃ v ∈ I, G.Adj u v ∧
    32 ∀ w ∈ S, G.Adj v w → w = u
    33
    34axiom value_eq {V : Type} [DecidableEq V] (G : SimpleGraph V)
    35 (S L R I : Finset V) (h : Partition G S L R I) :
    36 Grundy.value G S = g L.card R.card
    37
    38end Lax689614.Biclique
    39
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