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

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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 $n\ge1$ vertices, a plane cycle on $n\ge3$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 61 results over 20 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., Theorem 4.2.9 (standard reference, not scraped)

Euler's formula checked on a plane tree, a cycle, and the four-face embedding of K4K_4K4​