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.
A tournament is an orientation of a complete finite graph
Definition
A tournament is a finite digraph such that for every two distinct vertices , exactly one of and is an arc, and no loop is an arc. Equivalently, it is obtained by orienting every edge of the complete graph on (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
A tournament is strong when it is strongly connected in the sense of Directed walks, trails, paths and cycles, and strong connectivity. Digraph terminology follows Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph.
Depends on
Used by
- A transitive tournament has a directed Hamilton path and no directed cycle Example
- The directed 3-cycle is a strongly connected tournament with a directed Hamilton cycle Example
- A tournament vertex having an in-neighbour and an out-neighbour on a directed cycle can be inserted into that cycle Lemma
- Camion's theorem: every strongly connected tournament on at least three vertices has a directed Hamilton cycle Theorem
- Redei's theorem: every nonempty tournament has a directed Hamilton path Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Encyclopedia of Mathematics, Tournament (standard reference, not scraped)