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.
Holes, antiholes, and odd holes
Definition
A hole in a finite graph is an induced cycle of length at least (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges, Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
An antihole is the complement of a hole (Graph isomorphisms, automorphisms and graph complements). A hole or antihole is odd when it has an odd number of vertices.
Thus is both an odd hole and an odd antihole, because the complement of a -cycle is again a -cycle.
Depends on
Used by
- A split set in a bull-free graph Definition
- Basic and composite bull-free graphs Definition
- 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
- Strong Perfect Graph Theorem Remark
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
- Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 1 (standard reference, not scraped)