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.
Basic and composite bull-free graphs
Definition
A finite graph is composite when:
- is bull-free; and
- there exists an odd hole or odd antihole in such that some vertex of is complete to and some vertex of is anticomplete to .
A bull-free graph is basic when it is not composite.
Thus a basic bull-free graph forbids exactly the odd hole and odd antihole configurations that carry both a complete and an anticomplete outside witness.
Depends on
Used by
- Every prime bull-free graph is basic Corollary
- In a basic bull-free graph, an odd hole with a complete outside vertex has tightly constrained neighbors Lemma
- In a basic bull-free graph, an odd hole with an anticomplete outside vertex forbids consecutive neighbors Lemma
- Every basic bull-free graph is 2-narrow Theorem
- Every composite bull-free graph has a nontrivial module Theorem
- For a vertex in a basic bull-free graph, either its neighborhood or its antineighborhood is perfect Theorem
Dependency tree · two levels
7 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
- Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 1 (standard reference, not scraped)