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Transactions of the American Mathematical Society

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Ramsey theorems for knots, links and spatial graphs


Author: Seiya Negami
Journal: Trans. Amer. Math. Soc. 324 (1991), 527-541
MSC: Primary 57M25; Secondary 05C10
DOI: https://doi.org/10.1090/S0002-9947-1991-1069741-9
MathSciNet review: 1069741
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Abstract: An embedding $ f:G \to {{\mathbf{R}}^3}$ of a graph $ G$ into $ {{\mathbf{R}}^3}$ is said to be linear if each edge $ f(e)\quad (e \in E(G))$ is a straight line segment. It will be shown that for any knot or link type $ k$, there is a finite number $ R(k)$ such that every linear embedding of the complete graph $ {K_n}$ with at least $ R(k)$ vertices $ (n \geqslant R(k))$ in $ {{\mathbf{R}}^3}$ contains a knot or link equivalent to $ k$.


References [Enhancements On Off] (What's this?)

  • [BM] J. A. Bondy and U. S. R. Murty, Graph theory and its application, Macmillan, 1976.
  • [Br] A. F. Brown, Embedding of graphs in $ {E^3}$, Ph.D. Desertation, Kent State Univ., 1977.
  • [CF] R. H. Crowell and R. H. Fox, Introduction to knot theory, Springer-Verlag, 1963. MR 0146828 (26:4348)
  • [CG] J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), 445-453. MR 722061 (85d:57002)
  • [Fa] I. Fáry, On straight line representation of planar graphs, Acta Sci. Math. (Szeged) 11 (1948), 229-233. MR 0026311 (10:136f)
  • [GRS] R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey theory, Wiley, 1980. MR 591457 (82b:05001)
  • [Mo] T. H. Motzkin, Cooperative classes of finite sets in one and more dimensions, J. Combin. Theory 3 (1967), 244-251. MR 0214478 (35:5328)
  • [Ra] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (2) (1930), 264-286.
  • [Rb] D. F. Robinson, Symmetric embeddings of graphs, J. Combin. Theory 9 (1970), 377-400. MR 0278991 (43:4717)
  • [Ro] D. Rolfsen, Knots and links, Math. Lecture Series 7, Publish or Perish, 1976. MR 0515288 (58:24236)
  • [Sa] H. Sachs, On a spatial analogue of Kuratowski's theorem on planar graphs--an open problem, Graph Theory, Lagów 1981, Proceedings, Lecture Notes in Math., vol. 1018, Springer-Verlag, Berlin and Heidelberg, 1983, pp. 230-241. MR 730653 (85b:05077)
  • [St] S. K. Stein, Convex maps, Proc. Amer. Math. Soc. 2 (1951), 464-466. MR 0041425 (12:845h)
  • [Wa] K. Wagner, Bemerkungen zum Vierfarbenproblem, Jber. Deutsch. Math.-Verein. 46 (1936), 26-32.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1069741-9
Keywords: Knots, links, spatial graphs, Ramsey theory
Article copyright: © Copyright 1991 American Mathematical Society

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