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import FormalConjecturesUtilThe Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture
References:
[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
[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 SimpleGraphnamespace LovaszPlummerConjecturevariable {V : Type*}
The number of perfect matchings of G.
noncomputable def perfectMatchingCount (G : SimpleGraph V) : ℕ :=
{M : G.Subgraph | M.IsPerfectMatching}.ncardThe 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 := ⊢ ∃ c,
0 < c ∧
∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],
(∀ (v : V), G.degree v = 3) → G.IsBridgeless → 2 ^ (c * ↑(Fintype.card V)) ≤ ↑(perfectMatchingCount G)
All goals completed! 🐙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 := V:Typeinst✝²:Fintype Vinst✝¹:DecidableEq VG:SimpleGraph Vinst✝:DecidableRel G.Adjhcubic:∀ (v : V), G.degree v = 3hbridgeless:G.IsBridgeless⊢ 2 ^ (↑(Fintype.card V) / 3656) ≤ ↑(perfectMatchingCount G)
All goals completed! 🐙
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.supportSheehan'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 := ⊢ ∀ {V : Type} [inst : Fintype V] [inst_1 : DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],
(∀ (v : V), G.degree v = 4) →
∀ (v : V) (c : G.Walk v v),
IsHamiltonianCycle G c → ∃ w c', IsHamiltonianCycle G c' ∧ c'.edges.toFinset ≠ c.edges.toFinset
All goals completed! 🐙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 := V:Typeinst✝²:Fintype Vinst✝¹:DecidableEq VG:SimpleGraph Vinst✝:DecidableRel G.Adjr:ℕhr:300 ≤ rhreg:∀ (v : V), G.degree v = rv:Vc:G.Walk v vhc:IsHamiltonianCycle G c⊢ ∃ w c', IsHamiltonianCycle G c' ∧ c'.edges.toFinset ≠ c.edges.toFinset
All goals completed! 🐙end LovaszPlummerConjecture