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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
- Tournament property Sₖ: every set of at most k vertices is dominated by one vertex Definition
- 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
- Szele's bound: for every n≥1, some n-vertex tournament has at least n!/2ⁿ⁻¹ Hamilton paths Theorem
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
- Encyclopedia of Mathematics, Tournament (standard reference, not scraped)