Tokyo Journal of Mathematics

Intrinsically $n$-linked Complete Graphs

Gabriel C. DRUMMOND-COLE and Danielle O'DONNOL

Full-text: Open access


In this paper we examine the question: given $n>1$, find a function $f:\mathbf{N}\rightarrow \mathbf{N}$ 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 $\mathbf{R}^3$ 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.

Article information

Tokyo J. Math., Volume 32, Number 1 (2009), 113-125.

First available in Project Euclid: 7 August 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


DRUMMOND-COLE, Gabriel C.; O'DONNOL, Danielle. Intrinsically $n$-linked Complete Graphs. Tokyo J. Math. 32 (2009), no. 1, 113--125. doi:10.3836/tjm/1249648413.

Export citation


  • P. Blain, G. Bowlin, T. Fleming, J. Foisy, J. Hendricks and J. LaCombe, Some results on intrinsically knotted graphs, J. of Knot Theory Ramifications, 16(6) (2007), 749–760.
  • G. Bowlin and J. Foisy, Some new intrinsically 3-linked graphs, J. of Knot Theory Ramifications, 13(8) (2004), 1021–1027.
  • J. Conway and C. Gordan, Knots and links in spatial graphs, J. of Graph Theory, 7 (1983), 445–453.
  • E. Flapan, J. Foisy, R. Naimi and J. Pommersheim, Intrinsically n-linked graphs, J. of Knot Theory Ramifications, 10(8) (2001), 1143–1154.
  • E. Flapan, R. Naimi and J. Pommersheim, Intrinsically triple linked complete graphs, Topol. Appl., 115 (2001), 239–246.
  • D. O'Donnol, Intrinsically $n$-linked complete bipartite graphs, J. of Knot Theory Ramifications, 17(2) (2008), 133–139.
  • N. Robertson, P. Seymour and R. Thomas, Sachs' linkless embedding conjecture, J. of Combinatorial Theory, Series B, 64 (1995), 185–227.
  • H. Sachs, On spatial representations of finite graphs, Colloq. Math. Soc. János Bolyai, Vol. 37 (North-Holland, Budapest, 1984), 649–662.