CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A planar graph contains no subdivision of $K_5$ or $K_{3,3}$

Statement

A planar graph (Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs) contains no subgraph that is a subdivision of $K_5$ or $K_{3,3}$ in the sense of Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors.

Facts & Assumptions

Given: A planar graph $G$ .

[L1]

$K_5$ and $K_{3,3}$ are nonplanar ( $K_5$ and $K_{3,3}$ are nonplanar).

[F1]

A subdivision repeats edge subdivision zero or more times (Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors).

Proof

technique · contradiction

1.1

In a plane drawing of a subdivision, suppressing a degree-two subdivision vertex replaces its two incident polygonal edge arcs by their concatenation and preserves a plane drawing. Repeating this suppression shows that planarity of a subdivision implies planarity of the original graph.

2.1

Suppose $G$ contained a subdivision of $K_5$ or $K_{3,3}$ . A subgraph of a planar graph inherits a plane drawing, and step 1.1 would turn that drawing into a plane drawing of the corresponding original graph, contradicting [L1].

assume-contrastep 1.1L1F1discharge-contradiction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. Diestel, Graph Theory, 6th ed., Corollary 4.2.11 (standard reference, not scraped)

A planar graph contains no subdivision of K5K_5K5​ or K3,3K_{3,3}K3,3​