DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

A perfect graph

Definition

A finite simple graph $G$ is perfect when every induced subgraph $H$ of $G$ satisfies

$χ (H) = ω (H),$

where $χ (H)$ is the chromatic number and $ω (H)$ is the clique number (Proper vertex colourings and chromatic number, Cliques, stable sets, the clique number $ω (G)$ and stability number $α (G)$ , Subgraphs, induced subgraphs and spanning subgraphs).

Equivalently, every induced subgraph of a perfect graph can be coloured with as many colours as the size of one of its largest cliques, and no fewer.

Depends on

Used by

An α-narrow graph has a clique or stable set of size at least |V(G)|^1/(2α) Corollary
The five-cycle is bull-free but not perfect Counterexample
A good function on a graph Definition
Every perfect graph is 1-narrow Proposition
Strong Perfect Graph Theorem Remark
Substituting perfect graphs preserves perfection Remark
Weak Perfect Graph Theorem Remark
An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem
For a vertex in a basic bull-free graph, either its neighborhood or its antineighborhood is perfect 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 and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 2 (standard reference, not scraped)