ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Every odd cycle $C_{2 k + 1}$ has Turán density $1 / 2$

Example

For every $k \geq 1$ ,

$χ (C_{2 k + 1}) = 3 and π (C_{2 k + 1}) = \frac{1}{2} .$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

For $n \geq 3$ , $C_{n}$ has the consecutive edges ${i, i + 1}$ and the closing edge ${n - 1, 0}$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A proper $k$ -vertex-colouring is a map $c : V \to k$ with $c (u) \neq c (v)$ for every edge ${u, v}$ , its fibres are the colour classes, and $χ (G) = \min {k \in N : G is k -colourable}$ (Proper vertex colourings and chromatic number).

[F3]

For every finite graph $H$ with an edge, $π (H) = 1 - 1 / (χ (H) - 1)$ (The asymptotic extremal density is determined exactly by chromatic number: $π (H) = 1 - 1 / (χ (H) - 1)$ ).

Verification

technique · alternate colours and use odd parity

1.1

In a two-colouring of a cycle, colours must alternate along consecutive vertices. After the odd number $2 k + 1$ of edges, the closing edge would join equal colours, so no proper two-colouring exists.

givenF1F2

2.1

Colour vertices $0, \dots, 2 k - 1$ alternately with two colours and give vertex $2 k$ a third colour. This is proper, so $χ (C_{2 k + 1}) = 3$ . For $k = 1$ this is the triangle and the same argument applies.

step 1.1givenF1F2

3.1

The density formula gives $π (C_{2 k + 1}) = 1 - 1 / (3 - 1) = 1 / 2$ .

step 2.1givenF3∎

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: 38 results over 16 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

Yufei Zhao, Graph Theory and Additive Combinatorics (standard reference, not scraped)

Every odd cycle C2k+1C2k+1​ has Turán density 1/21/2