1. FormalConjectures
  2. ErdosProblems
  3. «762»
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Erdős Problem 762

References:

    erdosproblems.com/762

    [EGS90] Erdős, Paul and Gimbel, John and Straight, H. Joseph, Chromatic number versus cochromatic number in graphs with bounded clique number. European J. Combin. (1990), 235-240.

    [St24b] R. Steiner, On the difference between the chromatic and cochromatic number. arXiv:2408.02400 (2024).

namespace Erdos762

The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph.

Is it true that if $G$ has no $K_5$ and $\zeta(G)\geq 4$ then $\chi(G) \leq \zeta(G)+2$?

This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\omega(G)=4$, $\zeta(G)=4$, and $\chi(G)=7$.

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos762.lean"] theorem erdos_762 : answer(False) ↔ ∀ (V : Type*) [Fintype V] (G : SimpleGraph V), G.CliqueFree 5 → 4 ≤ G.cochromaticNumber → G.chromaticNumber ≤ G.cochromaticNumber + 2 := ⊢ False ↔ ∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V), G.CliqueFree 5 → 4 ≤ G.cochromaticNumber → G.chromaticNumber ≤ G.cochromaticNumber + 2 All goals completed! 🐙

A conjecture of Erdős, Gimbel, and Straight [EGS90], who proved that for every $n>2$ there exists some $f(n)$ such that if $G$ contains no clique on $n$ vertices then $\chi(G)\leq \zeta(G)+f(n)$.

@[category research solved, AMS 5] theorem erdos_762.variants.bounded_clique_number (n : ℕ) (hn : 2 < n) : ∃ f : ℕ, ∀ (V : Type*) [Fintype V] (G : SimpleGraph V), G.CliqueFree n → G.chromaticNumber ≤ G.cochromaticNumber + f := n:ℕhn:2 < n⊢ ∃ f, ∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V), G.CliqueFree n → G.chromaticNumber ≤ G.cochromaticNumber + ↑f All goals completed! 🐙

This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\omega(G)=4$, $\zeta(G)=4$, and $\chi(G)=7$.

@[category research solved, AMS 5] theorem erdos_762.variants.steiner : ∃ (n : ℕ) (G : SimpleGraph (Fin n)), G.cliqueNum = 4 ∧ G.cochromaticNumber = 4 ∧ G.chromaticNumber = 7 := ⊢ ∃ n G, G.cliqueNum = 4 ∧ G.cochromaticNumber = 4 ∧ G.chromaticNumber = 7 All goals completed! 🐙end Erdos762