ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The complete graph $K_n$ has chromatic number $n$

Example

For every $n\in\mathbb N$ , including $n=0$ , the complete graph $K_n$ satisfies

$\chi(K_n)=n.$

Facts & Assumptions

Given: The complete graph $K_n$ on the vertex set $n$ .

[L1]

Every two distinct vertices of $K_n$ are adjacent (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

[L2]

A proper colouring gives different colours to adjacent vertices, and $\chi$ is the least size of an available colour set (Proper vertex colourings and chromatic number).

Verification

technique · direct

1.1

In a proper colouring of $K_n$ , [L1] and [L2] force all vertices to receive distinct colours, so at least $n$ colours are required; this is also true at $n=0$ , where the assertion is vacuous.

L1L2

1.2

The identity function $n\to n$ is a proper $n$ -colouring because distinct vertices receive distinct values.

L1L2

2.1

The lower bound in step 1.1 and the colouring in step 1.2 give $\chi(K_n)=n$ .

step 1.1step 1.2L2∎

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: 27 results over 14 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

Springer, Graph Theory Fundamentals, Section 1.7.1 (standard reference, not scraped)

The complete graph KnK_nKn​ has chromatic number nnn