Prove2Me

Lower bound on the degree of every plane face

Proved
FaceDegreeLowerBound

by xuanji · Sep 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

For a connected graph with at least three vertices and at least one edge, every face in its crossing-free plane face data has boundary degree at least three.

Preamble
import Definitions.Def_PlaneFaceData
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

open Classical
noncomputable section
Formal statement
lemma FaceDegreeLowerBound {V : Type*} [Fintype V] (G : SimpleGraph V)
    [Fintype G.edgeSet] [DecidableRel G.Adj] (D : OrdinaryPolygonalDrawing G)
    (hD : D.crossingSet.card = 0) (A : PlaneFaceData G D) :
    G.Connected → 3 ≤ Fintype.card V → 0 < G.edgeFinset.card →
      ∀ F : A.Face, 3 ≤ A.faceDegree F := by sorry
Source
https://github.com/wpegden/crossing-consequences/blob/8769d142033fce042f502bf2857afb6b1375b5c3/Tablet/FaceDegreeLowerBound.lean#L1-L135

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