FormalConjectures.ErdosProblems.«914»

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import FormalConjecturesUtil

Erdős Problem 914

References:

erdosproblems.com/914

[CoHa63] Corrádi, K. and Hajnal, A., On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hungar. (1963), 423-439.

[HaSz70] Hajnal, A. and Szemerédi, E., Proof of a conjecture of P. Erdős. (1970), 601-623.

[KiKo08] Kierstead, H. A. and Kostochka, A. V., A short proof of the Hajnal-Szemerédi theorem on equitable colouring. Combin. Probab. Comput. (2008), 265-270.

namespace Erdos914

Let $r\geq 2$ and $m\geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$ contains $m$ vertex disjoint copies of $K_r$.

When $r=2$ this follows from Dirac's theorem. Corrádi and Hajnal [CoHa63] proved this when $r=3$. Hajnal and Szemerédi [HaSz70] proved this for all $r\geq 4$.

A shorter proof was given by Kierstead and Kostochka [KiKo08].

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos914.lean"]
theorem erdos_914 {r m : ℕ} (hr : 2 ≤ r) (hm : 1 ≤ m) {V : Type*} [Fintype V]
    (G : SimpleGraph V) [DecidableRel G.Adj] (hV : Fintype.card V = r * m)
    (hdeg : m * (r - 1) ≤ G.minDegree) :
    ∃ K : Fin m → Finset V, (∀ i, G.IsNClique r (K i)) ∧
      Pairwise fun i j => Disjoint (K i) (K j) := byr:ℕm:ℕhr:2 ≤ rhm:1 ≤ mV:Type u_1inst✝¹:Fintype VG:SimpleGraph Vinst✝:DecidableRel G.AdjhV:Fintype.card V = r * mhdeg:m * (r - 1) ≤ G.minDegree⊢ ∃ K, (∀ (i : Fin m), G.IsNClique r (K i)) ∧ Pairwise fun i j ↦ Disjoint (K i) (K j)
  sorryAll goals completed! 🐙

Equivalently, every graph with $rm$ vertices and maximum degree at most $m-1$ has a proper vertex colouring with $m$ colours in which every colour class has exactly $r$ vertices (an equitable colouring).

@[category research solved, AMS 5]
theorem erdos_914.variants.equitable_colouring {r m : ℕ} (hr : 2 ≤ r) (hm : 1 ≤ m) {V : Type*}
    [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] (hV : Fintype.card V = r * m)
    (hdeg : G.maxDegree ≤ m - 1) :
    ∃ C : G.Coloring (Fin m), ∀ i, (C.colorClass i).ncard = r := byr:ℕm:ℕhr:2 ≤ rhm:1 ≤ mV:Type u_1inst✝¹:Fintype VG:SimpleGraph Vinst✝:DecidableRel G.AdjhV:Fintype.card V = r * mhdeg:G.maxDegree ≤ m - 1⊢ ∃ C, ∀ (i : Fin m), (C.colorClass i).ncard = r
  sorryAll goals completed! 🐙

end Erdos914