A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway\â\\s conjecture states that a thrackle has at most as many edges as vertices. In this paper, we investigate the edge-vertex ratio of maximal thrackles, that is, thrackles in which no edge between already existing vertices can be inserted such that the resulting drawing remains a thrackle. For maximal geometric and topological thrackles, we show that the edge-vertex ratio can be arbitrarily small. When forbidding isolated vertices, the edge-vertex ratio of maximal geometric thrackles can be arbitrarily close to the natural lower bound of $12$. For maximal topological thrackles without isolated vertices, we present an infinite family with an edge-vertex ratio arbitrary close to~$ 45$.
|Titel||Graph Drawing and Network Visualization. GD 2019|
|Publikationsstatus||Veröffentlicht - 2019|
|Name||Lecture Notes in Computer Science (LNCS)|
Fields of Expertise
- Information, Communication & Computing
Aichholzer, O., Kleist, L., Klemz, B., Schröder, F., & Vogtenhuber, B. (2019). On the Edge-Vertex Ratio of Maximal Thrackles. in Graph Drawing and Network Visualization. GD 2019 (Band 11904, S. 482-495). (Lecture Notes in Computer Science (LNCS)). Prague, Czechia. https://doi.org/10.1007/978-3-030-35802-0_37, https://doi.org/10.1007/978-3-030-35802-0_37