FormalConjectures.ErdosProblems.«58»

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module

public import FormalConjecturesUtil

Erdős Problem 58

References:

erdosproblems.com/58

[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.

[GaHuMa21] Gao, Jun and Huo, Qingyi and Ma, Jie, A strengthening on odd cycles in graphs of given chromatic number. SIAM J. Discrete Math. (2021), 2317-2327.

[Gy92] Gyárfás, A., Graphs with k odd cycle lengths. Discrete Math. (1992), 41-48.

@[expose] public sectionnamespace Erdos58

If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains $K_{2k+2}$.

Conjectured by Bollobás and Erdős. Proved by Gyárfás [Gy92].

@[category research solved, AMS 5]
theorem erdos_58 :
    ∀ {V : Type*} (G : SimpleGraph V) (k : ℕ),
      G.oddCycleLengths.Finite →
      G.oddCycleLengths.ncard ≤ k →
        G.chromaticNumber ≤ (2 * k + 2 : ℕ∞) ∧
        (G.chromaticNumber = (2 * k + 2 : ℕ∞) ↔
          (SimpleGraph.completeGraph (Fin (2 * k + 2))).IsContained G) := by⊢ ∀ {V : Type u_1} (G : SimpleGraph V) (k : ℕ),
  G.oddCycleLengths.Finite →
    G.oddCycleLengths.ncard ≤ k →
      G.chromaticNumber ≤ 2 * ↑k + 2 ∧
        (G.chromaticNumber = 2 * ↑k + 2 ↔ (SimpleGraph.completeGraph (Fin (2 * k + 2))).IsContained G)
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos58