- 128 Downloads
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.
KeywordsGraph drawing Thrackle Outerplanar Alternating cycle Crossing rank
Unable to display preview. Download preview PDF.
- 7.Cairns, G., McIntyre, M., Nikolayevsky, Y.: The thrackle conjecture for K 5 and K 3,3. In: Towards a Theory of Geometric Graphs. Contemp. Math. vol. 342, pp. 35–54. American Mathematical Society, Providence, RI (2004)Google Scholar
- 11.Grünbaum, B.: Arrangements and spreads. In: Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 10. American Mathematical Society, Providence, RI (1972)Google Scholar
- 14.Pach, J.: Geometric graph theory. In: Surveys in combinatorics, 1999 (Canterbury), London Math. Soc. Lecture Note Ser., vol. 267, pp. 167–200. Cambridge University Press Cambridge (1999)Google Scholar
- 15.Pach, J., Agarwal Pankaj, K.: Combinatorial geometry. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1995)Google Scholar
- 16.Woodall, D.R.: Thrackles and deadlock. In: Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 335–347. Academic Press, London (1971)Google Scholar