Embedding products of graphs into Euclidean spaces

• Autorzy: Mikhail Skopenkov
• Języki publikacji: EN

Abstrakty

EN

For any collection of graphs $G₁,...,G_N$ we find the minimal dimension d such that the product $G₁ × ... × G_N$ is embeddable into $ℝ^d$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3,3})ⁿ$ are not embeddable into $ℝ^{2n}$, where K₅ and $K_{3,3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $Lk O → S^{2n-1}$, where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

Kategorie tematyczne

• 57Q35: Embeddings and immersions
• 57Q45: Knots and links (in high dimensions)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 179
• Numer: 3
• Strony: 191-198

Daty

wydano

2003

Twórcy

autor

Mikhail Skopenkov

• Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia

Identyfikatory

DOI

10.4064/fm179-3-1