Euler upper bound for drawable simple graphs

Open

WilliamsonShmoys.planar_edge_count_upper__p2m_23689463_5c07_4e0b_a29b_546de8e1e4a9_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Any finite drawable simple graph with at least three vertices has at most 3|V|−6 edges. This is the Euler upper-bound half of the target; a proof route is to establish Euler’s formula for the drawing’s faces and use that each face boundary has length at least three, accounting for disconnected drawings as well.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper__p2m_23689463_5c07_4e0b_a29b_546de8e1e4a9_a1 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (dL : DecidableRel L.Adj) (hL : HasPlanarDrawingOn L) (hcard : 3 <= Fintype.card V) : L.edgeFinset.card + 6 <= 3 * Fintype.card V := by sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

View graph