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c-closed graphs

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

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    Natural Language Statement

    Definition

    A graph is c-closed if every pair of distinct nonadjacent vertices has fewer than c common neighbors. The value c may be zero; exactly the complete graphs are 0-closed.

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    1import Mathlib.Combinatorics.SimpleGraph.Finite
    2import Mathlib.Data.Set.Card
    3
    4/-!
    5---
    6title: c-closed graphs
    7type: definition
    8---
    9A graph is c-closed if every pair of distinct nonadjacent vertices has
    10fewer than c common neighbors. The value c may be zero; exactly the
    11complete graphs are 0-closed.
    12-/
    13
    14namespace Lax560391.CClosedGraphs
    15
    16def IsCClosed {V : Type} [Fintype V] [DecidableEq V]
    17 (G : SimpleGraph V) (c : ℕ) : Prop :=
    18 ∀ u v : V, u ≠ v → ¬G.Adj u v →
    19 {w : V | G.Adj u w ∧ G.Adj v w}.ncard < c
    20
    21end Lax560391.CClosedGraphs
    22
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