ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The directed $3$ -cycle is a strongly connected tournament with a directed Hamilton cycle

Example

Let $T$ have vertices $1,2,3$ and arcs

1\to2,\qquad 2\to3,\qquad 3\to1.

Then $T$ is a strongly connected tournament, and $1,2,3,1$ is a directed Hamilton cycle.

Facts & Assumptions

Given: The three explicitly displayed arcs.

[F1]

A tournament has exactly one directed arc between each pair of distinct vertices (A tournament is an orientation of a complete finite graph).

[F2]

Strong connectivity requires a directed path for each ordered vertex pair, and a directed Hamilton cycle visits every vertex once before returning (Directed walks, trails, paths and cycles, and strong connectivity).

[L1]

Camion's theorem gives a directed Hamilton cycle in every strong tournament of order at least three (Camion's theorem: every strongly connected tournament on at least three vertices has a directed Hamilton cycle).

Verification

technique · direct

1.1

The three arcs orient the three unordered vertex pairs exactly once, so $T$ is a tournament. The ordered pairs $(1,2),(2,3),(3,1)$ are joined directly, while $(2,1),(3,2),(1,3)$ are joined by the two-arc paths through the remaining vertex. Hence $T$ is strongly connected.

givenF1F2

1.2

The closed list $1,2,3,1$ follows the three arcs and visits every vertex once before returning, so it is a directed Hamilton cycle, as [L1] predicts.

givenF2L1

2.1

This verifies all three assertions in the example.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 15 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)

The directed 333-cycle is a strongly connected tournament with a directed Hamilton cycle