ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Strong Perfect Graph Theorem, ‡ Substituting perfect graphs preserves perfection and ‡ Weak Perfect Graph Theorem. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The five-cycle is 2-narrow but not 1-narrow

Example

The cycle graph $C_{5}$ is two-narrow but not one-narrow.

Facts & Assumptions

Given: The cycle graph $C_{5}$ .

[L1]

Every bull-free graph is two-narrow (Every bull-free graph is 2-narrow).

[L2]

The graph $C_{5}$ is bull-free but not perfect (The five-cycle is bull-free but not perfect).

[F1]

A graph is one-narrow when every good function has total weight at most $1$ (An $α$ -narrow graph).

[F2]

A good function is a nonnegative weighting whose total on every perfect induced subgraph is at most $1$ (A good function on a graph).

Verification

technique · direct

1.1

By [L2], the graph $C_{5}$ is bull-free. The bull-free theorem [L1] therefore gives the two-narrow half of the example.

L1L2

1.2

Define $g (v) = 1 / 4$ for every vertex of $C_{5}$ . Every proper induced subgraph of $C_{5}$ is a forest on at most four vertices, hence is bipartite and has clique number at most $2$ ; the same is true for each of its induced subgraphs, so every proper induced subgraph of $C_{5}$ is perfect. Since [L2] says the whole $C_{5}$ is not perfect, the perfect induced subgraphs of $C_{5}$ are exactly the proper ones, and each has at most four vertices. Therefore [F2] makes $g$ a good function, because its total on any perfect induced subgraph is at most $4 \cdot (1 / 4) = 1$ . But $\sum_{v \in V (C_{5})} g (v) = 5 / 4 > 1$ , so [F1] shows that $C_{5}$ is not one-narrow.

L2F1F2algebra

2.1

Thus $C_{5}$ is two-narrow by step 1.1 but not one-narrow by step 1.2.

step 1.1step 1.2∎

Depends on

Used by

FALSE: every 2-narrow graph is 1-narrow False statement

Dependency tree · two levels

26 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, Sections 1-2 (standard reference, not scraped)