CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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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.
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$K_{3, 3}$ satisfies $∣ E ∣ \leq 3 ∣ V ∣ - 6$ but is nonplanar, so the planar edge bound is not sufficient

Statement refuted

Every finite simple graph with $n \geq 3$ vertices and at most $3 n - 6$ edges is planar.

Facts & Assumptions

Given: The complete bipartite graph $K_{3, 3}$ of Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices.

[L1]

Every simple planar graph with $n \geq 3$ vertices has at most $3 n - 6$ edges (Every simple planar graph with $n \geq 3$ vertices has at most $3 n - 6$ edges, with equality for every plane triangulation).

[L2]

The graph $K_{3, 3}$ is nonplanar ( $K_{5}$ and $K_{3, 3}$ are nonplanar).

Counterexample

technique · direct

1.1

The two parts of $K_{3, 3}$ contain three vertices each, so $∣ V ∣ = 6$ . Every vertex in either part is adjacent to all three vertices of the other part, giving $∣ E ∣ = 3 \cdot 3 = 9$ ; equivalently this follows from Handshake lemma: the sum of the vertex degrees is twice the number of edges. Consequently $9 \leq 3 \cdot 6 - 6 = 12$ , so the necessary inequality in [L1] holds.

L1algebra

2.1

Nevertheless [L2] says that this graph is nonplanar. Hence satisfying the planar simple-graph edge bound does not suffice for planarity.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

K3,3K3,3​ satisfies ∣E∣≤3∣V∣−6∣E∣≤3∣V∣−6 but is nonplanar, so the planar edge bound is not sufficient