CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

The five-cycle is bull-free but not perfect

Statement refuted

Every bull-free graph is perfect.

Facts & Assumptions

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

[F1]

The bull contains a triangle (The bull graph, A bull-free graph).

[F2]

The graph $C_{5}$ is the five-vertex cycle (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F3]

A graph is perfect when every induced subgraph satisfies $χ = ω$ (A perfect graph).

Counterexample

technique · direct

1.1

The graph $C_{5}$ is triangle-free by [F2], whereas every bull contains a triangle by [F1]. So $C_{5}$ contains no induced bull and is bull-free.

F1F2

1.2

In $C_{5}$ , the largest clique has size $2$ , while a proper vertex colouring needs $3$ colours. Hence $ω (C_{5}) = 2 < 3 = χ (C_{5})$ , so [F3] shows that $C_{5}$ is not perfect.

F2F3algebra

2.1

Therefore $C_{5}$ is a bull-free graph that is not perfect, refuting the claim.

step 1.1step 1.2∎

Depends on

Used by

The five-cycle is 2-narrow but not 1-narrow Example
FALSE: every bull-free graph is perfect False statement

Dependency tree · two levels

17 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)