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The Petersen graph has chromatic number $3$ , so its Turán density is $1 / 2$

Example

Let $P$ be the Petersen graph. Then

$χ (P) = 3 and π (P) = \frac{1}{2} .$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

In the Petersen graph's two-subset model, two vertices are adjacent exactly when the corresponding two-element subsets are disjoint (The Petersen graph on the two-element subsets of a five-element set, adjacent when disjoint).

[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 · exhibit an odd cycle and a three-colouring

1.1

The vertices $12, 34, 15, 23, 45$ form a $5$ -cycle in that order because consecutive pairs, including $45, 12$ , are disjoint. Hence $P$ is not bipartite and $χ (P) \geq 3$ .

givenF1F2

1.2

Partition the ten vertices into ${12, 13, 14, 15}$ , ${23, 24, 34}$ , and ${25, 35, 45}$ . Within each class every two subsets intersect, so the Petersen adjacency definition gives no edge within a class. This is a proper three-colouring, hence $χ (P) \leq 3$ .

givenF1F2

2.1

Steps 1.1-1.2 give $χ (P) = 3$ , and the density formula gives $π (P) = 1 - 1 / (3 - 1) = 1 / 2$ .

step 1.1step 1.2givenF3∎

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: 46 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)

The Petersen graph has chromatic number 33, so its Turán density is 1/21/2