FormalConjectures.ErdosProblems.«628»

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import FormalConjecturesUtil

Erdős Problem 628

References:

erdosproblems.com/628

[BKPS09] Balogh, József and Kostochka, Alexandr V. and Prince, Noah and Stiebitz, Michael, The Erdős-Lovász Tihany conjecture for quasi-line graphs. Discrete Math. (2009), 3985-3991.

[BrJu69] Brown, W. G. and Jung, H. A., On odd circuits in chromatic graphs. Acta Math. Acad. Sci. Hungar. (1969), 129-134.

[Er68b] Erdős, P., Problem 2. Theory of Graphs (1968), 361.

[So22] Song, Zi-Xia, A survey on the Erdős-Lovász Tihany conjecture. Adv. Math. (China) (2022), 259--274.

namespace Erdos628open SimpleGraph

Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\geq a$ and $\geq b$ respectively?

@[category research open, AMS 5]
theorem erdos_628 (V : Type*) [Fintype V] (G : SimpleGraph V) (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞) := byV:Type u_1inst✝:Fintype VG:SimpleGraph Vk:ℕhG_chrom:χ(G) = ↑khG_clique:G.CliqueFree ka:ℕb:ℕha:a ≥ 2hb:b ≥ 2hab:a + b = k + 1⊢ ∃ s, χ(induce s G) ≥ ↑a ∧ χ(induce sᶜ G) ≥ ↑b
  sorryAll goals completed! 🐙

Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles)..

@[category research solved, AMS 5]
theorem erdos_628.variants.k_5_a_3_b_3 (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG_chrom : G.chromaticNumber = (5 : ℕ∞))
    (hG_clique : G.CliqueFree 5) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (3 : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (3 : ℕ∞) := byV:Type u_1inst✝¹:Fintype VG:SimpleGraph Vinst✝:DecidableRel G.AdjhG_chrom:χ(G) = 5hG_clique:G.CliqueFree 5⊢ ∃ s, χ(induce s G) ≥ 3 ∧ χ(induce sᶜ G) ≥ 3
  sorryAll goals completed! 🐙

Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for quasi-line graphs.

@[category research solved, AMS 5]
theorem erdos_628.variants.quasi_line (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG_quasi : IsQuasiLineGraph G)
    (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞) := byV:Type u_1inst✝¹:Fintype VG:SimpleGraph Vinst✝:DecidableRel G.AdjhG_quasi:G.IsQuasiLineGraphk:ℕhG_chrom:χ(G) = ↑khG_clique:G.CliqueFree ka:ℕb:ℕha:a ≥ 2hb:b ≥ 2hab:a + b = k + 1⊢ ∃ s, χ(induce s G) ≥ ↑a ∧ χ(induce sᶜ G) ≥ ↑b
  sorryAll goals completed! 🐙

Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for graphs with independence number 2.

@[category research solved, AMS 5]
theorem erdos_628.variants.independence_number_2 (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG_indep : G.indepNum = 2)
    (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞) := byV:Type u_1inst✝¹:Fintype VG:SimpleGraph Vinst✝:DecidableRel G.AdjhG_indep:α(G) = 2k:ℕhG_chrom:χ(G) = ↑khG_clique:G.CliqueFree ka:ℕb:ℕha:a ≥ 2hb:b ≥ 2hab:a + b = k + 1⊢ ∃ s, χ(induce s G) ≥ ↑a ∧ χ(induce sᶜ G) ≥ ↑b
  sorryAll goals completed! 🐙

end Erdos628