Advertisement

GD 2012: Graph Drawing pp 346-351

# Tangles and Degenerate Tangles

• Andres J. Ruiz-Vargas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

## Abstract

We study some variants of Conway’s thrackle conjecture. A tangle is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a degenerate tangle. We settle a problem of Pach, Radoičić, and Tóth , by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles.

## Keywords

thrackles tangles degenerate tangles tangled thrackles graph drawing intersection graphs

## References

1. 1.
Unsolved problems. Chairman: P. Erdős. In: Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), Inst. Math. Appl., Southend-on-Sea, pp. 351–363 (1972)Google Scholar
2. 2.
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
3. 3.
Cairns, G., McIntyre, M., Nikolayevsky, Y.: The thrackle conjecture for $$K\sb 5$$ and $$K\sb{3,3}$$. In: Towards a Theory of Geometric Graphs, Contemp. Math., vol. 342, pp. 35–54. Amer. Math. Soc., Providence (2004)
4. 4.
Fulek, R., Pach, J.: A computational approach to Conway’s thrackle conjecture. Comput. Geom. 44(6-7), 345–355 (2011)
5. 5.
Kuratowski, K.: Sur le probleme des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)
6. 6.
Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18, 369–376 (1998)
7. 7.
Pach, J., Radoičić, R., Tóth, G.: Tangled thrackles. Geombinatorics (2012) (to appear)Google Scholar
8. 8.
Pach, J., Tóth, G.: Disjoint Edges in Topological Graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds.) IJCCGGT 2003. LNCS, vol. 3330, pp. 133–140. Springer, Heidelberg (2005)
9. 9.
Ringeisen, R.D.: Two old extremal graph drawing conjectures: progress and perspectives. Congressus Numerantium 115, 91–103 (1996)

## Copyright information

© Springer-Verlag Berlin Heidelberg 2013

## Authors and Affiliations

• Andres J. Ruiz-Vargas
• 1
1. 1.School of Basic SciencesÉcole Polytechnique Fédérale de LausanneSwitzerland

## Personalised recommendations

### Citepaper 