DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-09-09 (gpt-6-astra)

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 left six-vertex prime $H$ -graph

Definition

The left six-vertex prime $H$ -graph is the graph $L$ on vertices

${a, b, c, d, e, f}$

with edge set

${a b, b c, a d, b d, b e, d e, e f} .$

The labels follow the left drawing in Nguyen–Scott–Seymour, Figure 1: $a, b, c$ run from left to right along its top row and $d, e, f$ along its bottom row. The vertices $a, b, d, e$ form a diamond (a complete four-vertex graph with the edge $a e$ removed); $c$ is attached only to $b$ and $f$ only to $e$ . Their degrees in the displayed order are $2, 4, 1, 3, 3, 1$ .

Remarks

The earlier triangle-with-three-pendant-leaves description specified the net, a different graph with degree sequence $3, 3, 3, 1, 1, 1$ . It is superseded here; conclusions about that earlier graph do not establish conclusions about the Figure 1 graph.

Used by

Dependency tree · 0 levels

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

Sources

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. IV. New graphs with the Erdős-Hajnal property, Figure 1 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 2 (standard reference, not scraped)

The left six-vertex prime HH-graph