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Intrinsically \(n\)-linked complete graphs. (English) Zbl 1184.05026

Summary: In this paper we examine the question: given \(n>1\), find a function \(f:\mathbb N\rightarrow \mathbb N\) where \(m=f(n)\) is the smallest integer such that \(K_m\) is intrinsically \(n\)-linked. We prove that for \(n>1\), every embedding of \(K_{\lfloor \frac{7}{2}n\rfloor}\) in \(\mathbb R^3\) contains a non-splittable link of \(n\) components. We also prove an asymptotic result, that there exists a function \(f(n)\) such that \( \lim_{n\to \infty}\frac{f(n)}{n}=3\) and, for every \(n,\) \(K_{f(n)}\) is intrinsically \(n\)-linked.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C40 Connectivity
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
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References:

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