LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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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.

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

Statement

Let $c$ be a proper colouring, let $a \neq b$ , 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 $c (u) \neq c (v)$ 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 · two levels

11 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., Proposition 5.1.2 (standard reference, not scraped)