Vizing's conjecture (1968)

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import FormalConjecturesUtil

/-!
# Vizing's conjecture (1968)

*References:*
* [Wikipedia](https://en.wikipedia.org/wiki/Vizing%27s_conjecture)
* [Vi68] Vizing, V. G. (1968). "Some unsolved problems in graph theory."
  *Uspekhi Mat. Nauk* 23, pp. 117--134.
* [ClSu00] Clark, W. E. and Suen, S. (2000). "An inequality related to Vizing's conjecture."
  *Electron. J. Combin.* 7, N4.
* [SuTa12] Suen, S. and Tarr, J. (2012). "An improved inequality related to Vizing's conjecture."
  *Electron. J. Combin.* 19, P8.
* [BDGHHKR12] Brešar, B., Dorbec, P., Goddard, W., Hartnell, B. L., Henning, M. A., Klavžar, S.
  and Rall, D. F. (2012). "Vizing's conjecture: a survey and recent results."
  *J. Graph Theory* 69, pp. 46--76.
-/

open SimpleGraph

namespace VizingConjecture

/--
**Vizing's conjecture (1968).**

For all finite simple graphs $G$ and $H$, the domination number of the Cartesian (box)
product satisfies $\gamma(G \,\square\, H) \ge \gamma(G)\,\gamma(H)$.
-/
@[category research open, AMS 5]
theorem vizing_conjecture : answer(sorry) ↔
    ∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
      (G : SimpleGraph α) (H : SimpleGraph β),
      G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber := by
  sorry

/--
**The Clark–Suen inequality (2000).**

For all finite simple graphs $G$ and $H$,
$\gamma(G \,\square\, H) \ge \tfrac12\,\gamma(G)\,\gamma(H)$; that is, Vizing's conjecture holds
up to a factor of $2$.

*Reference:* [ClSu00].
-/
@[category research solved, AMS 5]
theorem vizing_conjecture.variants.clark_suen
    {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
    (G : SimpleGraph α) (H : SimpleGraph β) :
    G.dominationNumber * H.dominationNumber ≤ 2 * (G □ H).dominationNumber := by
  sorry

/--
**The Suen–Tarr inequality (2012).**

For all finite simple graphs $G$ and $H$,
$\gamma(G \,\square\, H) \ge \tfrac12\,\gamma(G)\,\gamma(H) +
\tfrac12\min\{\gamma(G), \gamma(H)\}$, improving the Clark–Suen bound.

*Reference:* [SuTa12].
-/
@[category research solved, AMS 5]
theorem vizing_conjecture.variants.suen_tarr
    {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
    (G : SimpleGraph α) (H : SimpleGraph β) :
    G.dominationNumber * H.dominationNumber + min G.dominationNumber H.dominationNumber ≤
      2 * (G □ H).dominationNumber := by
  sorry

/--
**Vizing's conjecture when `γ(H) = 1`.**

If `H` has a dominating vertex, so that $\gamma(H) = 1$, then Vizing's inequality reduces to
$\gamma(G) \le \gamma(G \,\square\, H)$, which is the projection bound
`dominationNumber_le_dominationNumber_boxProd`. This is the simplest of the known cases of the
conjecture (see [BDGHHKR12]).
-/
@[category research solved, AMS 5]
theorem vizing_conjecture.variants.dominationNumber_eq_one
    {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
    (G : SimpleGraph α) (H : SimpleGraph β) (hH : H.dominationNumber = 1) :
    G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber := by
  -- `γ(H) = 1` forces `H` to have a vertex: the empty graph has `γ = 0`.
  have hne : Nonempty β := by
    by_contra h
    rw [not_nonempty_iff] at h
    have : H.dominationNumber = 0 :=
      Nat.le_zero.mp (dominationNumber_le_of_isDominating H ∅ (fun v => (IsEmpty.false v).elim))
    omega
  rw [hH, mul_one]
  exact dominationNumber_le_dominationNumber_boxProd G H

end VizingConjecture

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