intrinsically linked

A graph Γ is intrinsically linkedMathworldPlanetmath if any embeddingMathworldPlanetmathPlanetmath of Γ in 3 contains a nontrivial link.

Example: K6, the complete graphMathworldPlanetmath on 6 vertices, was proven to be intrinsically linked by Conway and Gordon.

The property of being not intrinsically linked is inherited by minors. That is, if Γ is not intrinsically linked and Γ can be obtained from Γ by edge contractions or deletions, then Γ is also not intrinsically linked.
By the Robertson-Seymour Theorem (formerly Wagner’s Conjecture), the obstruction set for this property must be finite. This means that the set of minor minimalPlanetmathPlanetmath intrinsically linked graphs is finite. In fact, there are 7 graphs in this set; it is known as the Petersen family, and it consists of graphs which can be obtained from K6 by repeated -Y or Y- transformations.

Title intrinsically linked
Canonical name IntrinsicallyLinked
Date of creation 2013-03-22 18:06:11
Last modified on 2013-03-22 18:06:11
Owner YourInnerNurmo (20577)
Last modified by YourInnerNurmo (20577)
Numerical id 7
Author YourInnerNurmo (20577)
Entry type Definition
Classification msc 54J05
Related topic IntrinsicallyKnotted