LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

Swapping the two colours on one Kempe component preserves a proper colouring

Statement

Let $c$ be a proper colouring, let $a\ne b$ , and let $K$ be one $a$ - $b$ Kempe component (Kempe chains as connected components induced by two colour classes, Connected graphs and connected components defined by the existence of vertex paths). Interchanging $a$ and $b$ on $K$ and leaving all other colours fixed gives another proper colouring.

Facts & Assumptions

Given: The colouring $c$ , colours $a,b$ , and Kempe component $K$ .

[F1]

Properness means $c(u)\ne c(v)$ whenever $\{u,v\}\in E$ (Proper vertex colourings and chromatic number).

[F2]

A connected component is an induced subgraph on its maximal connected vertex set (Connected graphs and connected components defined by the existence of vertex paths).

Proof

technique · direct

1.1

Define $c'$ by swapping $a$ and $b$ at vertices of $K$ and setting $c'=c$ elsewhere.

F1F2

2.1

An edge with both endpoints in $K$ still has opposite $a,b$ colours after the swap, and an edge with neither endpoint in $K$ is unchanged. If exactly one endpoint lies in $K$ , the other endpoint cannot have colour $a$ or $b$ , for then that edge would place it in the same induced connected component $K$ . Its colour is therefore unaffected and differs from the swapped colour. Thus every edge remains proper.

step 1.1F1F2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 16 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., Proposition 5.1.2 (standard reference, not scraped)