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Outerplanar Thrackles

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Abstract

We show that a graph drawing is an outerplanar thrackle if and only if, up to an inversion in the plane, it is Reidemeister equivalent to an odd musquash. This establishes Conway’s thrackle conjecture for outerplanar thrackles. We also extend this result in two directions. First, we show that no pair of vertices of an outerplanar thrackle can be joined by an edge in such a way that the resulting graph drawing is a thrackle. Secondly, we introduce the notion of crossing rank; drawings with crossing rank 0 are generalizations of outerplanar drawings. We show that all thrackles of crossing rank 0 are outerplanar. We also introduce the notion of an alternating cycle drawing, and we show that a thrackled cycle is alternating if and only if it is outerplanar.

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Authors and Affiliations

  1. Department of Mathematics, La Trobe University, Melbourne, 3086, Australia

    Grant Cairns & Yury Nikolayevsky

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  1. Grant Cairns
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  2. Yury Nikolayevsky
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Correspondence to Grant Cairns.

Additional information

We thank J. Pach and T. Zaslavsky for their valuable comments. We also thank the referee, whose comments and suggestions improved the presentation of the paper.

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Cairns, G., Nikolayevsky, Y. Outerplanar Thrackles. Graphs and Combinatorics 28, 85–96 (2012). https://doi.org/10.1007/s00373-010-1010-1

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  • DOI: https://doi.org/10.1007/s00373-010-1010-1

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