Alphabeta MathHow 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.
Which small graphs count as prime on this page
This page adopts the direct module-theoretic convention: a graph is prime when its only modules are the trivial ones. Under that convention every graph on one or two vertices is prime, and no graph on exactly three vertices is prime (No graph on exactly three vertices is prime).
Some sources build a size restriction into the terminology instead. Chudnovsky phrases primality through non-substitutability for graphs with at least two vertices, while other texts reserve the word prime for graphs on at least four vertices. These conventions agree with the direct module-theoretic convention on graphs with at least four vertices, but deliberately differ at smaller orders. The equivalence with non-substitutability is stated with its size hypotheses in A graph with at least two vertices is prime exactly when it is not obtained by substituting one graph on at least two vertices for a vertex of another graph on at least two vertices.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
13 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
- M. Habib and C. Paul, A Survey on Algorithmic Aspects of Modular Decomposition, sec. 2.4 (standard reference, not scraped)
- M. Chudnovsky, The Erdős–Hajnal Conjecture — A Survey, sec. 2 (standard reference, not scraped)
- Y. Huang, Q. Ju, and X. Zhou, Erdős-Hajnal beyond the five-vertex path, sec. 1.2 (standard reference, not scraped)