LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 star-expansion of the four-vertex path contains induced six- and seven-cycles

Statement

Let $P_{4}^{⋆}$ be the star-expansion of the path $b_{1} - b_{2} - b_{3} - b_{4}$ . Then $P_{4}^{⋆}$ contains induced copies of $C_{6}$ and $C_{7}$ .

Facts & Assumptions

Given: The star-expansion $P_{4}^{⋆}$ with root $v$ , teeth $a_{1}, a_{2}, a_{3}, a_{4}$ , and path vertices $b_{1}, b_{2}, b_{3}, b_{4}$ .

[L1]

In a star-expansion, the only new edges are $v a_{i}$ and $a_{i} b_{i}$ (The star-expansion of a graph).

Proof

technique · direct finite check

1.1

Consider the six vertices $v, a_{1}, b_{1}, b_{2}, b_{3}, a_{3}$ . By [L1], the edges among them are exactly $v a_{1}, a_{1} b_{1}, b_{1} b_{2}, b_{2} b_{3}, b_{3} a_{3}, a_{3} v$ . No other edge is present: the teeth $a_{1}, a_{3}$ meet only their matched path vertices and the root, and the path contains no edge $b_{1} b_{3}$ . Hence these six vertices induce a $6$ -cycle.

1.2

Consider the seven vertices $v, a_{1}, b_{1}, b_{2}, b_{3}, b_{4}, a_{4}$ . Again [L1] shows that the edges among them are exactly $v a_{1}, a_{1} b_{1}, b_{1} b_{2}, b_{2} b_{3}, b_{3} b_{4}, b_{4} a_{4}, a_{4} v$ . No chord occurs: $a_{1}$ and $a_{4}$ meet only the root and their matched path vertices, and the path has no edges other than the consecutive ones displayed. Hence these seven vertices induce a $7$ -cycle.

2.1

Steps 1.1 and 1.2 give induced copies of $C_{6}$ and $C_{7}$ in $P_{4}^{⋆}$ .

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

8 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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, discussion after Theorem 1.9 (standard reference, not scraped)