FormalConjectures.Paper.ReedOmegaDeltaChi

FormalConjectures
Paper
ReedOmegaDeltaChi

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import FormalConjecturesUtil

Reed's omega, delta, and chi conjecture

References:

B. Reed, ω Δ and χ, J. Graph Theory 27 (1998) 177-212.

openproblemgarden

mathoverflow/37923 asked by user Andrew D. King


open Classical

open scoped Finset


namespace ReedOmegaDeltaChi

For a graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the chromatic number. Reed's omega, delta, and chi conjecture states that $$\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$$

@[category research open, AMS 5]
theorem declaration uses 'sorry'reed_omega_delta_chi_conjecture :
  ∀ {V : Type} (G : SimpleGraph V),
    let χ := G.chromaticNumber
    let ω := G.ecliqueNum
    let Δ := G.emaxDegree
    2 * χ ≤ ω + Δ + 2 := by⊢ ∀ {V : Type} (G : SimpleGraph V),
  let χ := G.chromaticNumber;
  let ω := G.ecliqueNum;
  let Δ := G.emaxDegree;
  2 * χ ≤ ω + Δ + 2
  sorryAll goals completed! 🐙

For a finite graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the chromatic number. Reed's omega, delta, and chi conjecture states that $$\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$$

@[category research open, AMS 5]
theorem declaration uses 'sorry'reed_omega_delta_chi_conjecture_for_finite_graphs :
  ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
    let χ := G.chromaticNumber
    let ω := G.cliqueNum
    let Δ := G.maxDegree
    2 * χ ≤ ω + Δ + 2 := by⊢ ∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],
  let χ := G.chromaticNumber;
  let ω := G.cliqueNum;
  let Δ := G.maxDegree;
  2 * χ ≤ ↑ω + ↑Δ + 2
  sorryAll goals completed! 🐙

The simplest open case is when $\Delta(G) = 6$ and $\omega(G) = 2$.

@[category research open, AMS 5]
theorem declaration uses 'sorry'reed_conjecture_Δ_6_ω_2 :
  ∀ {V : Type} (G : SimpleGraph V), G.emaxDegree = 6 ∧ G.cliqueNum = 2 → G.chromaticNumber ≤ 5 := by⊢ ∀ {V : Type} (G : SimpleGraph V), G.emaxDegree = 6 ∧ G.cliqueNum = 2 → G.chromaticNumber ≤ 5
    sorryAll goals completed! 🐙


end ReedOmegaDeltaChi