Alphabeta MathHow 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.
Perfect graphs
Definition
A finite graph is perfect when every induced subgraph of satisfies
where is the chromatic number and is the clique number (Proper vertex colourings and chromatic number, Cliques, stable sets, the clique number and stability number , Subgraphs, induced subgraphs and spanning subgraphs).
Under the library conventions, the null graph is perfect because its unique induced subgraph is itself and .
Depends on
Used by
- The five-vertex path is perfect but not a cograph Example
- A complete connection of two perfect graphs is perfect Lemma
- A disjoint union of two perfect graphs is perfect Lemma
- The perfect-induced-subgraph formulation of the Erdos-Hajnal conjecture Remark
- A pure blockade with a perfect pattern has a large complete or anticomplete subblockade Theorem
- Every cograph is perfect Theorem
- Every perfect graph satisfies |V(G)|<=kappa(G) Theorem
- The Erdos-Hajnal property is equivalent to the large-cograph, large-perfect, and kappa formulations Theorem
Dependency tree · two levels
10 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
- Maria Chudnovsky, The Erdos-Hajnal Conjecture - A Survey, Introduction (standard reference, not scraped)