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Kempe chains as connected components induced by two colour classes
Definition
Let be a proper vertex colouring of a graph (Proper vertex colourings and chromatic number) and let be colours. The - Kempe subgraph is the subgraph induced by the vertices whose colours lie in (Subgraphs, induced subgraphs and spanning subgraphs). An - Kempe chain is a connected component of this induced subgraph (Connected graphs and connected components defined by the existence of vertex paths).
The word chain denotes a connected component, not necessarily a graph-theoretic path. A path inside a Kempe chain alternates colours because the ambient colouring is proper.
Depends on
Used by
- A degree-five vertex is inserted into a plane graph after one explicit Kempe-chain colour swap Example
- For five cyclically ordered neighbours of a plane vertex, alternating Kempe paths between the first and third and between the second and fourth cannot both occur Lemma
- Swapping the two colours on one Kempe component preserves a proper colouring Lemma
- Five colour theorem: every planar graph has chromatic number at most five Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 28 results over 15 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)