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Edge clique covers

Lax560391.EdgeCliqueCovering · concepts/Lax560391/EdgeCliqueCovering.lean · lax-560391

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    Natural Language Statement

    Definition

    An edge clique cover is an indexed family of vertex cliques covering every edge. Cliques may overlap, be empty, or repeat; isolated vertices need not be covered.

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    1import Mathlib.Combinatorics.SimpleGraph.Clique
    2
    3/-!
    4---
    5title: Edge clique covers
    6type: definition
    7---
    8An edge clique cover is an indexed family of vertex cliques covering every
    9edge. Cliques may overlap, be empty, or repeat; isolated vertices need not
    10be covered.
    11-/
    12
    13namespace Lax560391.EdgeCliqueCovering
    14
    15structure EdgeCliqueCovering {V : Type} [DecidableEq V]
    16 (G : SimpleGraph V) (k : ℕ) where
    17 cliques : Fin k → Finset V
    18 isClique : ∀ i, ∀ u ∈ cliques i, ∀ v ∈ cliques i, u ≠ v → G.Adj u v
    19 covers : ∀ ⦃u v⦄, G.Adj u v → ∃ i, u ∈ cliques i ∧ v ∈ cliques i
    20
    21end Lax560391.EdgeCliqueCovering
    22
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    From Mathlib

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