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Lax10.Theorem2

Theorem 2

concepts/Lax10/Theorem2.lean · lax-10

proven

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    Theorem

    Let m≥4m \geq 4. If a finite simple graph has chromatic number at least m+1m + 1, then it has a 33-connected subgraph with chromatic number at least mm.

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    1import Lax10.ThreeConnectedAndColorable
    2
    3/-!
    4---
    5title: Theorem 2
    6type: theorem
    7---
    8Let m≥4m \geq 4. If a finite simple graph has chromatic number at least
    9m+1m + 1, then it has a 33-connected subgraph with chromatic number at least
    10mm.
    11-/
    12
    13namespace Lax10.Theorem2
    14
    15universe u
    16
    17open Lax10.ThreeConnectedAndColorable
    18
    19/--
    20If χ(G)≥m+1\chi(G) \geq m + 1, some 33-connected subgraph of GG has chromatic
    21number at least mm.
    22-/
    23axiom theorem2 {V : Type u} [Fintype V] [DecidableEq V]
    24 (m : Nat) (G : SimpleGraph V)
    25 (hm : 4 ≤ m) (hchi : ((m + 1 : Nat) : ℕ∞) ≤ G.chromaticNumber) :
    26 ∃ H : G.Subgraph, ThreeConnected H.coe ∧
    27 (m : ℕ∞) ≤ H.coe.chromaticNumber
    28
    29end Lax10.Theorem2
    30
    Show Proof

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