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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Linkless embeddings of graphs in $3$-space
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by Neil Robertson, P. D. Seymour and Robin Thomas PDF
Bull. Amer. Math. Soc. 28 (1993), 84-89 Request permission

Abstract:

We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the complement in 3-space of the embedding of every subgraph is free. (ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of ${K_5}$ or ${K_{3,3}}$. (iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by "3-switches", an analog of 2-switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other. (iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from ${K_6}$ by means of $Y \Delta$- and $\Delta Y$-exchanges.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 28 (1993), 84-89
  • MSC: Primary 57M15; Secondary 05C10
  • DOI: https://doi.org/10.1090/S0273-0979-1993-00335-5
  • MathSciNet review: 1164063