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Lax10.Fragile

Fragile and m-fragile graphs

concepts/Lax10/Fragile.lean · lax-10

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    Definition

    An mm-fragile graph is a graph whose 33-connected subgraphs are all (m−1)(m - 1)-colorable. A fragile graph is a graph with no 33-connected subgraph.

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    1import Lax10.ThreeConnectedAndColorable
    2
    3/-!
    4---
    5title: Fragile and m-fragile graphs
    6type: definition
    7---
    8An mm-fragile graph is a graph whose 33-connected subgraphs are all
    9(m−1)(m - 1)-colorable. A fragile graph is a graph with no 33-connected
    10subgraph.
    11-/
    12
    13namespace Lax10.Fragile
    14
    15universe u
    16
    17variable {V : Type u}
    18
    19open Lax10.ThreeConnectedAndColorable
    20
    21/--
    22An mm-fragile graph is one whose 33-connected subgraphs are all
    23(m−1)(m - 1)-colorable.
    24-/
    25def MFragile (m : Nat) (G : SimpleGraph V) : Prop :=
    26 ∀ H : G.Subgraph, ThreeConnected H.coe → KColorable (m - 1) H.coe
    27
    28/-- A graph is fragile when none of its subgraphs is 33-connected. -/
    29def HasNoThreeConnectedSubgraph (G : SimpleGraph V) : Prop :=
    30 ∀ H : G.Subgraph, ¬ ThreeConnected H.coe
    31
    32end Lax10.Fragile
    33

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