ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 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.

$H_{1}$ and $H_{5}$ arise by the stated labelled leaf attachments

Example

The recursive definition of $H_{0}, \dots, H_{5}$ produces the intermediate graph $H_{1}$ and the final graph $H_{5}$ exactly by the labelled leaf attachments.

Facts & Assumptions

Given: The recursive family $H_{0}, H_{1}, \dots, H_{5}$ .

[L1]

The graph $H_{1}$ is obtained from $H_{0}$ by adjoining a leaf $v_{1}^{'}$ at $v_{1}$ , and for each $i = 1, \dots, 5$ , the graph $H_{i}$ is obtained from $H_{i - 1}$ by adjoining a leaf $v_{i}^{'}$ at $v_{i}$ (The graphs $H_{0}, H_{1}, \dots, H_{5}$ ).

Verification

technique · direct finite check

1.1

By [L1], $H_{1}$ keeps all edges of $H_{0}$ and adds exactly one new edge $v_{1} v_{1}^{'}$ . So $H_{1}$ is precisely the five-wheel with one pendant leaf at $v_{1}$ .

2.1

Repeating the same operation for $i = 2, 3, 4, 5$ adds the leaves $v_{2}^{'}, v_{3}^{'}, v_{4}^{'}, v_{5}^{'}$ one at a time and changes no earlier adjacencies. Hence $H_{5}$ is the graph obtained from $H_{0}$ by attaching one leaf to each rim vertex $v_{i}$ .

step 1.1L1∎

Depends on

The graphs $H_0,H_1,\ldots,H_5$

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 7 (standard reference, not scraped)

H1H1​ and H5H5​ arise by the stated labelled leaf attachments