ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-03

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.

The complete graph $K_{n}$ has chromatic number $n$

Example

For every $n \in N$ , including $n = 0$ , the complete graph $K_{n}$ satisfies

$χ (K_{n}) = n .$

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 $χ$ 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 $χ (K_{n}) = n$ .

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

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

The complete graph KnKn​ has chromatic number nn