TheoremStatement: Literature-sourcedProof: AI-adaptedSession-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.

Every bull-free graph is 2-narrow

Statement

Every bull-free finite graph is two-narrow.

Facts & Assumptions

Given: A bull-free finite graph $G$ .

[L2]

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

[L3]

Every composite bull-free graph has a nontrivial module (Every composite bull-free graph has a nontrivial module).

[L4]

Graphs defined by forbidden induced subgraphs form hereditary classes (Every class defined by forbidden induced subgraphs is hereditary, $H$ -free and $F$ -free graphs under the induced-subgraph convention).

[L5]

Substitution preserves $α$ -narrowness, hence in particular two-narrowness (Substituting two $α$ -narrow graphs yields another $α$ -narrow graph).

[F1]

A module is a vertex set whose outside vertices are each complete or anticomplete to it (Modules of a graph, and the trivial modules).

[F2]

The substitution $H_{1} [v \to H_{2}]$ replaces the vertex $v$ by the graph $H_{2}$ and gives every vertex of $H_{2}$ exactly the outside adjacencies of $v$ (Substituting one graph for a vertex of another).

Proof

technique · direct

1.1

We argue by induction on $∣ V (G) ∣$ . If $G$ is basic, then [L2] proves the claim. So assume that $G$ is not basic. Because “basic” means “bull-free and not composite”, the bull-free graph $G$ is then composite, and [L3] gives a nontrivial module $X$ . Choose $x \in X$ , let $H_{2} = G [X]$ , and let $H_{1} = G [(V (G) ∖ X) \cup {x}]$ . Since $X$ is nontrivial and proper, both $H_{1}$ and $H_{2}$ have fewer vertices than $G$ . By [L4], both are bull-free because they are induced subgraphs of $G$ .

L2L3L4F1chooseinduction

2.1

Because $X$ is a module, every vertex outside $X$ is complete or anticomplete to $X$ . Therefore [F2] shows that $G$ is exactly the substitution $H_{1} [x \to H_{2}]$ . By the inductive hypothesis, both $H_{1}$ and $H_{2}$ are two-narrow, so [L5] makes $G$ two-narrow as well.

step 1.1F1F2L5

3.1

Either $G$ was basic, when step 1.1 reduced directly to [L2], or it was composite, when step 2.1 proved it two-narrow. Hence every bull-free finite graph is two-narrow.

step 1.1step 2.1L2∎

Depends on

Used by

Dependency tree · two levels

36 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 Erdős-Hajnal Conjecture: A Survey, Theorem 2.4 (standard reference, not scraped)
Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Theorem 1.3 (standard reference, not scraped)