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.
The right six-vertex prime -graph is the complement of the left one, and is prime
Example
The right six-vertex prime -graph is the complement of the left one, and is prime.
Facts & Assumptions
Given: The left and right six-vertex prime -graphs on the common label set .
The right graph is defined as the complement of the left graph (The right six-vertex prime -graph, The left six-vertex prime -graph, Graph isomorphisms, automorphisms and graph complements).
The left six-vertex prime -graph is prime (The left six-vertex prime -graph is prime, and deleting any pendant leaf gives the bull).
A graph is prime exactly when it has no nontrivial module (Prime graphs: those whose only modules are the trivial ones, Modules of a graph, and the trivial modules).
A vertex set is a module of a graph if and only if it is a module of the complement, because outside vertices swap complete and anticomplete behaviour.
Verification
By [L1], the identity map on the common label set is an isomorphism from the right graph to the complement of the left graph.
Since the left graph is prime by [L2], [L3] says it has no nontrivial module. By [F1], its complement also has no nontrivial module. Therefore the right graph is prime by [L3].
Thus the right six-vertex prime -graph is the complement of the left one and is prime.
Depends on
- The right six-vertex prime $\mathcal H$-graph
- The left six-vertex prime $\mathcal H$-graph
- Prime graphs: those whose only modules are the trivial ones
- Modules of a graph, and the trivial modules
- Graph isomorphisms, automorphisms and graph complements
- The left six-vertex prime $\mathcal H$-graph is prime, and deleting any pendant leaf gives the bull
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 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
- 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)