The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture

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import FormalConjecturesUtil

/-!
# The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture

*References:*
* [Wikipedia](https://en.wikipedia.org/wiki/Petersen%27s_theorem#Related_conjectures)
* [LP86] Lovász, L. and Plummer, M. D. (1986). *Matching Theory.* North-Holland.
* [EKKKN11] Esperet, L., Kardoš, F., King, A. D., Král', D. and Norine, S. (2011).
  "Exponentially many perfect matchings in cubic graphs." *Adv. Math.* 227, pp. 1646--1664.
  [arXiv:1012.2878](https://arxiv.org/abs/1012.2878)
* [Sh77] Sheehan, J. (1977). "The multiplicity of Hamiltonian circuits in a graph." In *Recent
  Advances in Graph Theory*, Academia, Prague, pp. 477--480.
* [Th98] Thomassen, C. (1998). "Independent dominating sets and a second Hamiltonian cycle in
  regular graphs." *J. Combin. Theory Ser. B* 72, pp. 104--109.
-/

open SimpleGraph

namespace LovaszPlummerConjecture

variable {V : Type*}

/-- The number of perfect matchings of `G`. -/
noncomputable def perfectMatchingCount (G : SimpleGraph V) : ℕ :=
  {M : G.Subgraph | M.IsPerfectMatching}.ncard

/--
**The Lovász–Plummer conjecture (1970s), proved by Esperet, Kardoš, King, Král' and Norine
(2011).**

Every bridgeless cubic graph on $n$ vertices has exponentially many perfect matchings: there is
a constant $c > 0$ such that the number of perfect matchings is at least $2^{cn}$.
[EKKKN11] prove this with $2^{n/3656}$.
-/
@[category research solved, AMS 5]
theorem lovasz_plummer_conjecture :
    ∃ c : ℝ, 0 < c ∧ ∀ {V : Type} [Fintype V] [DecidableEq V]
      (G : SimpleGraph V) [DecidableRel G.Adj],
      (∀ v, G.degree v = 3) → G.IsBridgeless →
      (2 : ℝ) ^ (c * Fintype.card V) ≤ perfectMatchingCount G := by
  sorry

/--
**The explicit bound of Esperet–Kardoš–King–Král'–Norine (2011).**

Every bridgeless cubic graph on $n$ vertices has at least $2^{n/3656}$ perfect matchings.

*Reference:* [EKKKN11].
-/
@[category research solved, AMS 5]
theorem lovasz_plummer_conjecture.variants.explicit
    {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (hcubic : ∀ v, G.degree v = 3) (hbridgeless : G.IsBridgeless) :
    (2 : ℝ) ^ ((Fintype.card V : ℝ) / 3656) ≤ perfectMatchingCount G := by
  sorry

/-- A **Hamiltonian cycle** of `G`: a cycle passing through every vertex. -/
def IsHamiltonianCycle (G : SimpleGraph V) {v : V} (c : G.Walk v v) : Prop :=
  c.IsCycle ∧ ∀ w, w ∈ c.support

/--
**Sheehan's conjecture (1977).**

Every $4$-regular graph with a Hamiltonian cycle has a second Hamiltonian cycle (one with a
different edge set). Sheehan's conjecture would settle the last open case of the question,
raised by Smith's theorem for cubic graphs, of which regular Hamiltonian graphs have a second
Hamiltonian cycle: Thomassen [Th98] proved it for all $r$-regular graphs with $r \ge 300$.
-/
@[category research open, AMS 5]
theorem sheehan_conjecture :
    ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
      (∀ v, G.degree v = 4) →
      ∀ (v : V) (c : G.Walk v v), IsHamiltonianCycle G c →
        ∃ (w : V) (c' : G.Walk w w), IsHamiltonianCycle G c' ∧
          c'.edges.toFinset ≠ c.edges.toFinset := by
  sorry

/--
**Thomassen (1998): regular graphs of large degree.**

Every $r$-regular Hamiltonian graph with $r \ge 300$ has a second Hamiltonian cycle.

*Reference:* [Th98].
-/
@[category research solved, AMS 5]
theorem sheehan_conjecture.variants.thomassen
    {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj]
    (r : ℕ) (hr : 300 ≤ r) (hreg : ∀ v, G.degree v = r)
    (v : V) (c : G.Walk v v) (hc : IsHamiltonianCycle G c) :
    ∃ (w : V) (c' : G.Walk w w), IsHamiltonianCycle G c' ∧
      c'.edges.toFinset ≠ c.edges.toFinset := by
  sorry

end LovaszPlummerConjecture

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