Edge bound for a graph with a crossing-free ordinary drawing

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PlanarEdgeBound

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

graph-theoryplanar-graphs

Let $G$ be a finite simple graph. If $G$ admits an ordinary polygonal drawing whose crossing set has cardinality zero, then its number of edges is at most three times its number of vertices:

|E(G)|\le 3|V(G)|.

This is the global planar edge bound used after removing one selected edge for each crossing point. It is stated for the drawing formalism used by the crossing-number development and includes disconnected graphs.

Preamble

import Definitions.Def_OrdinaryPolygonalDrawing

open Classical
noncomputable section

Formal statement

lemma PlanarEdgeBound {V : Type*} [Fintype V] (G : SimpleGraph V) [Fintype G.edgeSet] :
    (∃ D : OrdinaryPolygonalDrawing G, D.crossingSet.card = 0) →
      G.edgeFinset.card ≤ 3 * Fintype.card V := by sorry

Source

https://github.com/wpegden/crossing-consequences/blob/8769d142033fce042f502bf2857afb6b1375b5c3/Tablet/PlanarEdgeBound.lean#L1-L64

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