DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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 graphs $H^{+}$ and $H^{-}$ for two distinguished vertices

Definition

Let $H$ be a finite graph and let $u, v \in V (H)$ be distinct.

The graph $H^{+}$ is obtained from $H$ by adjoining a new vertex $w$ adjacent to both $u$ and $v$ , and also adjoining the edge $u v$ when it is not already present.

The graph $H^{-}$ is obtained from $H$ by adjoining a new vertex $w$ adjacent to both $u$ and $v$ , and deleting the edge $u v$ when it is present.

Thus $H^{+}$ forces the distinguished pair $u, v$ to be adjacent, while $H^{-}$ forces it to be nonadjacent, and in both cases the new vertex $w$ is adjacent exactly to $u$ and $v$ .

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Section 1.4 (standard reference, not scraped)

The graphs H+H+ and H−H− for two distinguished vertices

Definition

Used by

Dependency tree · 0 levels

Sources

The graphs $H^{+}$ and $H^{-}$ for two distinguished vertices