ReedOmegaDeltaChi.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 3 contain sorry, 0 declared as axioms

Reed's omega, delta, and chi conjecture

References:

Imports (1)

FormalConjecturesUtil

theorem ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture 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 sorry line 41 source
Source documentation
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 $$\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$$

depends on sorry what this means
Statement uses: SimpleGraph, ENat, SimpleGraph.chromaticNumber, SimpleGraph.ecliqueNum, SimpleGraph.emaxDegree, LE.le, HMul.hMul, OfNat.ofNat, Nat, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem reed_omega_delta_chi_conjecture :
  ∀ {V : Type} (G : SimpleGraph V),
    let χ := G.chromaticNumber
…
```
theorem ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture_for_finite_graphs 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 sorry line 55 source
Source documentation
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 $$\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$$

depends on sorry what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, ENat, SimpleGraph.chromaticNumber, Nat, SimpleGraph.cliqueNum, SimpleGraph.maxDegree, LE.le, HMul.hMul, OfNat.ofNat, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem reed_omega_delta_chi_conjecture_for_finite_graphs :
  ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
    let χ := G.chromaticNumber
…
```
theorem ReedOmegaDeltaChi.reed_conjecture_Δ_6_ω_2 The simplest open case is when $\Delta(G) = 6$ and $\omega(G) = 2$ sorry line 67 source
Source documentation
The simplest open case is when $\Delta(G) = 6$ and $\omega(G) = 2$.

depends on sorry what this means
Statement uses: SimpleGraph, And, Eq, ENat, SimpleGraph.emaxDegree, OfNat.ofNat, Nat, SimpleGraph.cliqueNum, LE.le, SimpleGraph.chromaticNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem reed_conjecture_Δ_6_ω_2 :
  ∀ {V : Type} (G : SimpleGraph V), G.emaxDegree = 6 ∧ G.cliqueNum = 2 → G.chromaticNumber ≤ 5 := by
…
```