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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
namespace OAI
namespace Barnette
universe u
/-- A topological plane drawing of a simple graph, with no crossings. -/
structure PlaneEmbedding {V : Type u} (G : SimpleGraph V) where
point : V → ℝ × ℝ
point_injective : Function.Injective point
arc : ∀ (v w : V), G.Adj v w → Path (point v) (point w)
arc_injective : ∀ (v w : V) (h : G.Adj v w),
Function.Injective (arc v w h)
arc_reverse : ∀ (v w : V) (h : G.Adj v w),
arc w v h.symm = (arc v w h).symm
arc_avoids_vertices : ∀ (v w : V) (h : G.Adj v w)
(t : unitInterval) (z : V),
arc v w h t = point z →
(t = 0 ∧ z = v) ∨ (t = 1 ∧ z = w)
interiors_disjoint : ∀ (v w x y : V) (h : G.Adj v w) (k : G.Adj x y)
(t s : unitInterval),
t ≠ 0 → t ≠ 1 → s ≠ 0 → s ≠ 1 →
arc v w h t = arc x y k s →
(v = x ∧ w = y) ∨ (v = y ∧ w = x)
/-- Planarity means existence of a crossing-free topological plane embedding. -/
def Planar {V : Type u} (G : SimpleGraph V) : Prop :=
Nonempty (PlaneEmbedding G)
/-- Deleting any set of at most two vertices leaves a connected graph of order at least two. -/
def ThreeVertexConnected {V : Type u} [Fintype V] (G : SimpleGraph V) : Prop :=
4 ≤ Fintype.card V ∧
∀ S : Finset V, S.card ≤ 2 → (G.induce {v | v ∉ S}).Connected
/-- A single spanning cycle, not a possibly disconnected spanning 2-factor. -/
def HasHamiltonianCycle {V : Type u} [DecidableEq V] (G : SimpleGraph V) : Prop :=
∃ v : V, ∃ p : G.Walk v v, p.IsHamiltonianCycle
/-- Cubic bipartite three-vertex-connected plane graphs have a Hamiltonian cycle. -/
def MainStatement : Prop :=
∀ (V : Type u) [Fintype V] [DecidableEq V] (G : SimpleGraph V)
[DecidableRel G.Adj],
G.IsRegularOfDegree 3 → G.IsBipartite → Planar G →
ThreeVertexConnected G → HasHamiltonianCycle G
end Barnette
end OAI
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