ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

Euler's formula checked on a plane tree, a cycle, and the four-face embedding of $K_{4}$

Example

Euler's formula gives the same value on a plane tree, a cycle, and the tetrahedral embedding of $K_{4}$ .

Facts & Assumptions

Given: A plane tree on $n \geq 1$ vertices, a plane cycle on $n \geq 3$ vertices, and the standard plane embedding of $K_{4}$ .

[L1]

Every connected plane graph satisfies $∣ V ∣ - ∣ E ∣ + ∣ F ∣ = 2$ (Euler's formula $∣ V ∣ - ∣ E ∣ + ∣ F ∣ = 2$ for every connected plane graph).

[L2]

A finite forest satisfies $∣ V ∣ = ∣ E ∣ + c$ , so a tree on $n$ vertices has $n - 1$ edges (For every forest, $∣ V ∣ = ∣ E ∣ + c$ , where $c$ is the number of connected components).

Verification

technique · direct

1.1

By [L2], a tree on $n$ vertices has $n - 1$ edges, and its plane embedding has one face by Every plane forest has exactly one face. Thus $∣ V ∣ - ∣ E ∣ + ∣ F ∣ = n - (n - 1) + 1 = 2$ . This includes the one-vertex tree, for which the edge count is zero.

L1L2algebra

1.2

A plane cycle $C_{n}$ , in the notation of Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices, has $n$ vertices and $n$ edges. Its polygon bounds one face and has the unbounded face on the other side, so $∣ V ∣ - ∣ E ∣ + ∣ F ∣ = n - n + 2 = 2$ .

L1algebra

2.1

Embed three vertices of $K_{4}$ as a triangle and place the fourth inside it, joined to all three corners. The graph has four vertices, six edges, three bounded triangular faces, and the unbounded triangular face. Hence $∣ V ∣ - ∣ E ∣ + ∣ F ∣ = 4 - 6 + 4 = 2$ , as [L1] requires.

L1constructalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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., Theorem 4.2.9 (standard reference, not scraped)

Euler's formula checked on a plane tree, a cycle, and the four-face embedding of K4K4​